CORRELATION OF HEAT PIPE PARAMETERS

A THESIS

Presented to

The Faculty of the Division.of Graduate

Studies and. Research

" . ' • • b y

Colquitt Lamar Williams

In Partial Fulfillment

of the Requirements for the Degree

Doctor of Philosophy in the

School of Mechanical Engineering

Georgia Institute of Technology

March, 1973

CORRELATION OF HEAT PIPE PARAMETERS

Approved:

/)

~?—^xT^—— " ! ~

G. T. Golwell, Chairman

'-*\

N. Zuber '

H. Z. Black

//J. H. Rust: r^ ,

1 . — .

H. C. Ward 1 -) & 7 :?

Date Approved by Chairman: ^ • •-••-u' / —>

11

ACKNOWLEDGMENTS

The author gratefully acknowledges the assistance of his thesis

advisor, Dr. G-. T. Cdwell, for continual patience and instructive guid

ance. The author's initial and continued interests in heat pipes

stemmed from Dr. Colwell's encouragements.

The author is indebted to Drs._ N. Zuber,/ J. H. Rust, H. C. Ward,

W. Z. Black, and A. E. Bergles for their creative suggestions and con

structive criticisms in determining thesis scope and methods. Each

professor served diligently on the thesis advisory committee or reading

committee.

The author extends special thanks to Dr. S. P. Kezios, Director,

School of Mechanical Engineering, for providing financial aid with the

EDEA Fellowship and graduate teaching assistantships. Special thanks

to Mr. L. A. Cavalli for his help and interests in fabricating the heat

pipe systems.

TABLE OF COMMENTS

P a g e

ACIOTOWLEDGMENTS i i

LIST OF TABLES v i i

LIST OF ILLUSTRATIONS v i i i

NOMENCLATURE xi

SUMMARY xvii

Chapter

I. INTRODUCTION l

1. Relevance of the Prohlem 2. Advances Required 3« Description of the Work

A. Analytical B. Experimental Ci Correlation

- II. RELATED INVESTIGATIONS k

1. Operating Characteristics

A. Theoretical B. Experimental C. Comparison of Theories to Experiments

2. Parameter Studies

A. Fluid Properties B. Geometric Factors k

C. Environmental Coupling

III. GENERAL THEORY 32

1. Objective 2. Overall Concepts 3. Reference Parameters 4. Vapor Region

TABLE OF CONTENTS (Continued)

Page

A. Spatial Boundaries B. Assumptions C. Field Equations in Vector Form D. Constitutive Equations in General Form E. Dimensionless Groups

5. Liquid Region

• A.' Spatial .Boundaries B. Assumptions C. Field Equations in Vector Form D. Constitutive Equations in General Form E. Dimensionless Groups

6. Liquid-Vapor Interface

A. Spatial. Location B. Assumptions C. Boundary Conditions in Vector Form D. Constitutive Equations in General Form E. Dimensionless Groups

7. Environmental Boundary Conditions

A. Spatial Location B. Assumptions" • . C. Boundary Conditions in Vector Form D. Constitutive Equations in General Form E. Dimensionless Groups

8. Heat Pipe Operating Temperature 9. Summary of Dimensionless Groups

EXPERIMEMTATION

1. Objective 2. Equipment

A. Heat Pipe B. Auxiliary Equipment

3. Procedure

• • 59

r

A. Heat Pipe Construction B. Preliminary Testing

TABLE OF CONTENTS (Continued)

Chapter Page

C. Start Up D. Measurements E. Change of Operating Point •F. Shut Down G. Terminal Testing

k. Results

A. Direct Data B. Data Reduction C. Reduced Data

V. CORRELATION THEORY . 80

1. Objective 2. Overall System 3. Condenser k. Evaporator 5. System of Equations

A. Dimensional Equations B. Dimensionless Equations C. Dimensionless Groups D. Dimensionless Equations in Terms of

TT Symbols

6. Results and Discussion

A. Solution of Correlation Equations B. Comparison of Experimental Values to

Predicted Values

VI. CONCLUSIONS AND RECOMMENDATIONS 12k

1. Conclusions 2. Recommendations

Appendices

A. GENERAL' THEORY EQUATIONS IN DIMENSIONLESS FORM . . . 127

B. VACUUM LEAK TESTING 150

C. EVALUATION OF FLUID INJECTION VOLUME 153

D. OPERATING DATA 157

vi

TABLE OF CONTENTS (Continued)

Appendices Page

E. PRESSURE AND TEMPERATURE EMF DATA 158

F. PRESSURE TRANSDUCER DATA REDUCTION 165

G. POTENTIOMETER DATA REDUCTION 166

H. RECORDER DATA REDUCTION . 168

I. HEAT TRANSFER RATE DATA REDUCTION 170

J. HEAT SINK TEMPERATURE DATA REDUCTION 179

K. CONDENSER EOTRNAL UNIT CONDUCTANCE DATA REDUCTION . . 180

L. ESTIMATION OF INTERFACIAL INERTIAL FORCES . 182

M. ESTIMATION OF VAPOR PRESSURE DROP . . . . . . . . . . . 18^

N. EVALUATION OF WICK FRICTION FACTOR l86

0. ESTIMATION OF LIQUID INERTIA TERM 189

P. ESTIMATION OF MAGNITUDE OF THERMAL RESISTANCES . . . . 192

Q. EVALUATION OF CONDENSER ACTIVE LENGTH FROM MEASURED t PARAMETERS 195

R. DERIVATION OF EQUATION FOR EFFECTIVE WICK THERMAL CONDUCTIVITY . . . . . . . . . . . . . 198

S. EVALUATION OF EVAPORATOR ACTIVE LENGTH FROM MEASURED PARAMETERS 203

T. DERIVATION OF RELATION-FOR EVAPORATOR EFFECTIVE WICK THICKNESS 211

U. EVALUATION OF INTERNAL THERMAL RESISTANCE .' 21^

V. DATA AND THEORIES OF OTHER INVESTIGATORS . 2l8

BiB.LiO(mAPffir::.::. 22^

VITA 228

LIST OF TABLES

Page

Initially Identified Heat Pipe Parameters 19

General Theory Reference Relations . 35

Dimensionless Groups of General Theory 56

Geometric Parameters of Experimental Heat Pipe 6k

Thermocouple Materials 66

Experimental Equipment List 68

Reduced Operating Data . . 77

Axial Profile Data Sampling for Water . 78

Axial Profile Data Siampling for Methanol 79

Correlation Dimensionless Group Listing 107

LIST OF ILLUSTRATIONS

viii

Figure Page

1. Cotter's Heat Pipe Schematic 6

2. Comparison of Predicted to Measured Values of Maximum Heat Rates ... . . . . . . . 17

3. Heat Pipe Fluid Parameter Versus Temperature for Various Fluids 23

k. Qmwi Versus Vv/rw for Various Values of V'Ki 25

5. Variation of K> with Mass Flux Rate 26

6. Comparison of Predicted to Measured Values of Wick Effective Thermal Conductivity . . 29

7. General Theory Heat Pipe Schematic . 33

8. General Theory Spatial Boundaries 1+0

9. Experimental Heat Pipe Dimensions 60

10. Experimental Heat Pipe Condenser Header 6l

11. Experimental Heat Pipe Instrument Locations 62

12. Experimental Auxiliary Equipment Schematic 63

13. Typical Axial Temperature Distributions 80

Ik. Correlation Theory Heat Pipe Schematic 82

15. Correlation Theory Thermal Resistances 85

16. Correlation Model of Condenser Active Length 87

17. Correlation Data Curve for Condenser Active Length -r ;~, . Ratio and Axial Vapor Reynolds Number . . . '. 90

18. Condenser Liquid-Vapor Interface Schematic . . 91

19. Condenser Thermal Resistance Schematic 92

20. Correlation Model of Evaporator Active Length . . . . ' . . 95

ix

LIST OF ILLUSTRATIONS (Continued)

Figure Page

21. Pressure Balance Loop at End of Active Evaporator $6

22. Correlation Data Curve for Evaporator Active Length Ratio and Meniscus Radius Ratio . . . 98

23. Evaporator Effective Wick Thickness Schematic . . . . . . . 100

2k. Evaporator Effective Wick Thickness Ratio Versus Meniscus Radius Ratio . . . . 101

25. Predicted Values of Condenser Active Length Ratio Versus Axial Vapor Reynolds Number Ill

26. Predicted Values of Evaporator, Active Length Ratio Versus Heat Pipe Number . . . . . . . . . . . 112

27. Predicted Values of Evaporator Meniscus Radius Ratio Versus Heat Pipe Number 113

28. Predicted Values of Evaporator Effective Wick Thickness Versus Heat Pipe Number Il4

29. Predicted Values of Wick Effective Thermal Conductivity-Ratio Versus Wick Solid-Liquid Thermal Conductivity-Ratio 116

30. Predicted Values of Heat Pipe Operating Temperature Ratio Versus Condenser Heat Sink Number . 117

31. Comparison of Predicted and Measured Values of Heat Pipe Operating Temperature Ratio 118

32. Comparison of Predicted and Measured Values of Condenser Active Length Ratio ., 119

33. Comparison of Predicted and Measured Values of Evaporator Active Length Ratio ., 120

3^. Comparison of Predicted and Measured Values of Overall Heat Pipe Thermal Resistance . 121

7-B-l Vacuum Leak Test 152

C-l Schematic Diagram for Estimation of Wick Porosity 15k

G-l Cooling Water Temperature Difference Versus EMF 167

}

. X

I

LIST OF ILLUSTRATIONS (Continued)

Figure Page

H-l Recorder EMF Fraction Versus Temperature 169

I-l Cooling Water Heat Capacity Versus Temperature 172

N-l Wick Cross Section for Evaluation of Wick Friction Factor . 187

R-1 Jfeomeibr y .Model; f o r ; Conduct i1on;:;TJteqaghl iDne ;jjayer of Screen . . . . . . . . . . 199

R-2 Annular Cross Section for Conduction Through Screen Layers with Liquid Gap . . 201

S-l Shell Diagram for Evaporator 20 -

T-l Interface Model for Derivation of 212

V-l Power Versus Evaporator Wick Temperature . ;.. ; , Difference for Data of Fox, et al. (10) 223

xi

NOMENCLATURE

Latin

°- detailed geometry surface parameter

A area

Aw total cross sectional wick area

^u effective, wick cross sectional area for liquid flow

k width of wick for rectangular heat pipes

&A. Biot no. defined iDy Eq. (II. k o )

*-? specific heat at constant pressure

Ec Eckert no.

u A.°* function, nondimensional

»x •/•* function, dimensional

^ gravitational acceleration

h specific enthalpy

hje liquid specific enthalpy as averaged over wick cross section

hvA latent heat ;of vaporization

H unit conductance "between outer shell wall and surroundings

^w wick specific permeattility

K wick friction factor (inverse permea"bility)

K» reference value of wick friction factor"

J? total heat pipe length

i* design length of adia"batic section

X<- design length of condenser section

ie. design length of evaporator section

XI1

NOMENCLATURE (Continued)

Latin

"4jb effective frictional length defined by Eq. (V. k)

t~c active length of condenser

Le active length of evaporator

*** mass flow rate

M dimensionless group defined "by Eq. (II. 39)

v\fti unit normal vectors on liquid-vapor interface, see Figure

fi*,^t unit normal vectors.on pipe "boundary, see Figure 8

n number of layers of mesh screen

P pressure

-f> surface coordinate defined "by Eq. (ill. 66)

<x surface coordinate defined "by Eq. (ill. 6j)

^ conduction heat flux vector defined "by Eq. (ill. 15)

Q heat transfer rate

r radial coordinate from heat pipe centerline

fc wick pore radius

Vn» evaporator meniscus radius

fiww minimum evaporator meniscus radius

Tp outer radius of pipe shell

T* radial coordinate on liquid-vapor interface,

ft vapor space radius from heat pipe centerline

TIN outer radius of wick

fos wick-solid radius, one half wire diameter for mesh screen

P>« Reynolds number

Xlll

K$MEICLATURE (Continued)

Latin

R*» variable meniscus radius

R**» evaporator wick thermal resistance

P-cvx condenser wick thermal resistance

R*x evaporator interface thermal resistance

Rci. condenser interface thermal resistance

Rv vapor region thermal resistance

5 vapor flow parameter used in Eq. (II. 8)

t^tx unit tangent vectors

t . tortuosity used in Eq. (II. k)

T temperature

"Tc condenser sink temperature

T"n evaporator source temperature.

To heat pipe operating temperature

Tp outer pipe shell surface temperature

"Tew condenser average temperature at outer wick surface

Tew evaporator average temperature at outer wick surface

U unit conductance «

Uc unit conductance between outer wick surface and condenser sink

UH unit conductance between outer wick surfaces and evaporator source

u radial velocity

a* vapor reference radial velocity defined by Eq. (ill. 2)

uji liquid reference radial, velocity defined by Eq. (ill. 3)

*°" angular component of velocity

xiv

roiMCIATUKE (Continued)

Latin

V velocity vector

VJL inter facial velocity vector

Vr radial component of velocity

*+> axial velocity

y axial coordinate

Greek

tf vapor radial Reynolds no. defined "by Eq. (II. l6)

<5 wick thickness

Se evaporator effective wick thickness

fc wick porosity

*) radial coordinate

Is radial coordinate, ^a1(rJe)

B angular coordinate, % - TjCT'ft)

X thermal conductivity

X ? pipe shell thermal conductivity

X«LL effective liquid-wick thermal,conductivity

Xs wick solid thermal conductivity

X second coefficient of viscosity

M. absolute viscosity

TT number, 3. 1^159 • • •

it* -*th dimensionless group

f> density

NOMENCLATURE (Continued)

surface tension

fluid friction tensor

fluid contact angle

condenser

condenser pipe

evaporator

effective'

environment

heat source

indice

indice

indice

liquid

liquid-condenser

liquid-evaporator

reference value

pipe

radial component

surface; surroundings

vapor

vapor-condenser

xvi

NOMENCLATURE (Continued)

Subscripts

" e vapor-evaporator

w s wick-solid

•^ axial component

& angular component

v radial component

Superscripts and Overscripts• .

* ratio of model to prototype

6 average value of variable G

c* unit vector in G direction

c» vector of variable G

-v- limit from above

- limit from below

Special Operators

V gradient

V' divergence

7s surface gradient

%' surface divergence

dot product

NOTE: Symbols which cannot be found here are defined in the respective Appendices.

XVI1

SUMMARY

An analytical and experimental study is conducted for the purpose

of identifying and correlating heat pipe parameters. These parameters

are studied under three topics, with each section providing new contri

butions toward the understanding of heat pipe performance.

Dimensional analysis is applied to a general theory, which yields

the first nondimensional writing of the detailed field and constitutive

equations for the heat pipe vappr and liquid regions. In addition to

these equations, boundary conditions at the liquid-vapor interface and

the environment are studied. This overall study results in a new list

ing of twenty four potentially important dimensionless groups. Included

is one previously unidentified group which contains the radius of the

wick solid (one half wire diameter for mesh screen).

Experiments are conducted on a horizontal, stainless steel heat

pipe, with the wick composed of two layers of mesh screen separated by

a thin liquid film. The tests use working fluids of water and methanol

for a data range of power, 20 - 350 Btu/hr, and operating temperature,

k-2 - 73 F. Other data are collected for use in evaluating the param

eters required for correlation. Independent parameters, those extern

nally controlled during the tests, are determined to be the condenser

sink temperature, unit conductance between outer wick wall and condenser

sink, and power delivered to the evaporator.

A correlation theory is analysed for simplification of the general

theory. This correlation theory yields a system of solvable equations

XVI11

for heat pipe performance and identification of significant dimension-

less groups. Included in the twenty groups are active length ratios of

heat pipe condenser and evaporator sections.

The length along which condensation occurs may differ from the

design length, depending on the vapor's axial Reynolds number. The

length along which evaporation occurs may differ from the design length,

depending on the evaporator meniscus radius. These newly identified

dependencies and the solved correlation equations yield predictions of

the dependent parameters for cases previously untested, experimental

values observed in this investigation, and experimental values from the

works of others. Comparisons of correlation theory predictions to

experimental values indicate agreement to within 10 per cent.

CHAPTER I

INTRODUCTION

1. Relevance of the Problem

The use of heat pipes in source-sink couplings is of interest to

various technologies "because many engineering operations require devices

with high effective thermal conductivities. Examples of such systems

include gas turbine "blades, spacecraft radiators, nuclear reactors,

electronic components, heat exchangers, and thermionic converters.

In order to design a heat pipe for a particular application, all

significant parameters governing heat pipe performance must first be

identified and then interpretations made as to which variables are in

dependent, i.e. free variables which the designer may select. Examples

of the significant parameters include working fluid properties, geomet

rical factors, operating temperatures, heat transfer rates, and the

areas through which heat flows. A failure to consider all parameters

may result in inadequate designs.

2. Advances Required

From the discussion of related investigations in Chapter II it

can be concluded that the following advances are required:

1) Establish the listing of significant heat pipe variables and

a corresponding list of dimensionless groups.

2) Obtain mathematical models which can correctly predict heat

pipe performance below the wieking limit.

2

3) Present results in a parametric form which are useful to

researchers and designers.

h) Corroberate the predicted results with experimental data.

3. Description of the Work

The purpose of this investigation is to study the interrelations

"between the parameters governing heat pipe performance. Goals are

divided into three parts.

A. Analytical

Analytical goals are

1) Write the conservation equations of mass, momentum, and

energy for the dynamic's of vapor and liquid flow in a heat pipe operating

with a single working fluid,.

2) Write the conservation equations of mass, momentum, and

energy for the liquid-vapor interface and external boundary conditions.

3) Write the necessary constitutive equations.

k) Obtain a listing of dimensionless groups from these equations.

B. Experimental

Experimental goals are

1) Operate a single horizontal heat pipe of fixed geometry first

with water as the working fluid and then with methanol.

2) Collect performance data necessary for evaluation of the

dimensionless groups required for correlation of the groups.

C. Correlation

Correlation goals are

l) Obtain mathematical models which yield the simplified gov

erning equations for heat pipe operation.

2) Indicate the important dimensionless groups.

3) Provide interpretations for distinguishing "between inde

pendent and dependent variables.

k) Provide predicted solutions of the governing relations for

previously untested combinations of heat pipe variables.

5) Compare predicted values to experimental values obtained in

this study and the works of others.

k

CHAPTER II

RELATED INVESTIGATIONS

The heat pipe concept came into open literature in 1964 when

Grover, et al. (.11)*, published experimental results. Grover demon

strated that a heat pipe could he successfully operated with working

fluids of water and sodium. Since Grover's work, many investigations

have "been conducted on the heat pipe. Source listings and reviews are

found in the works of Cheung (4), Winter and Borsch (38) j and Shinnick

(32). Analytical and experimental investigations directly related to

this study concern the topics of Operating Characteristics and Parameter

Studies.

1. Operating Characteristics

Understanding the general heat pipe operating characteristics

consists of describing the internal dynamics of energy and mass flow

and the corresponding limitations. Analytical and experimental studies

have "been conducted.

A. Theoretical

The work of Cotter (8) provided the first attempt to analyze the

heat pipe. Cotter initiated a quantitative engineering theory for the

design and performance analysis of heat pipes. Cotter's model, see

^Numbers in parentheses correspond to references listed in the Bibliography.

5

Figure 1, consists of a simple heat pipe inclined in a gravity field

and having only evaporator and condenser sections, i.e. no adia"batic

section.

Cotter expressed conservation of mass in steady state "by

y.(fv^ * o (II.1)

for both liquid and vapor regions. Taking the velocity of both fluid

phases as zero at the evaporator end of the heat pipe, Cotter integrated

Equation (il.l) to obtain

-rrw(V) +• i*fc(V) = ° (II-2)

where »v and mi are;the net mass flow rates in the axial direction of

the vapor and liquid regions respectively.

Conservation of momentum in steady state was expressed "by

7p = rf M ? ' H ) - PV-(7V) (II.3)

while assuming constant viscosity and density for each fluid phase.

These assumptions were suitable since Cotter defined the heat pipe

operation to take place with small changes in temperature and estimated

vapor Mach numbers were low.

For liquid flow, Cotter attempted to simplify Equation (II.3) by

1) averaging the terms over a cross sectional area of the wick with

dimensions small compared to the wick thickness and shell dimensions

but large compared to the average radius of a pore, rt and 2) using

simple order of magnitude analyses to indicate negligible inertia terms

and radial pressure gradient terms. Under these approximations Cotter

^GRAVITY

Figure 1. Cotter's Model.

obtained the expression

7

^ - p.asA^e - >x**^ ,TT M

where t and £ are dimensionless numbers characterizing the wick geom

etry by tortuosity and porosity respectively. Cotter referred to

Equation (II.4) as a "version of Darcy's law". This was reasonable

since Darcy's law for relating the pressure gradient to mass flow rate

for flow in a porous media is (27)

dP _ M± (II. 5)

where Kw is the wick permeability and /Vs. is the effective flow area of

the liquid.

In order to estimate the pressure gradient in the vapor region,

Cotter modeled this flow as being dynamically similar to pipe flow with

injection or suction through a porous wall. Cotter used the results of

Yuan and Finkelstein (39) and Knight and Mclnteer (15) who studied the

problem under the basic assumptions of'l) laminar flow, 2) steady state,

3) constant fluid properties, k) two dimensional flow, and 5) uniform

wall injection or suction velocity. The assumption of uniform wall

velocity implies that the wall Reynolds number^, 2f; defined by

• c M7~ ~~ T.TTA>/ T y ( I I . D )

i s a constant.

8

For the v i scous dominated regime of I tf l^l Cot te r recommended

the express ion (39)

T> " IT***^ * Y OO * ;. CII.T)

This relation had "been obtained "by a perturbation solution of the

momentum and continuity equations under the stated assumptions. In this

expression R/ is-interpreted as the pressure of the vapor as averaged

over the cross -section of the heat pipe.

For high evaporation or condensation rates, |Y( » \

Cotter recommended the expression (15)

^ "' " wtfTfr / (I1*8)

where 5 •= \ for evaporation and S = +/TT'L for condensation.

This relation had been obtained from an exact solution of the momentum

and continuity equations.

Cotter expressed the conservation of energy by the relation

*'% '- ° > (II.9)

where

^ = V\ p v - xvT (II. 10)

where V\ and X are the specific enthalpy and thermal conductivity of

the working fluid respectively. Cotter stated that an energy equation,

formed "by substituting Equation 11.10 into II.9* "was subject to the

assumptions of l) steady state, 2) no energy sources, and 3) negligible

heat exchange by radiation. Cotter's energy equation implies that the

work done by pressure and viscous actions are negligible, which is

reasonable since low velocities are presumed to be encountered and the

corresponding parameter EU/Re . (3) is much less than unity for

typical heat pipes.

Cotter proceeded by defining the total axial heat transfer rate,

Q (V); of the heat pipe by

&VV--XVV^r^ , (Il.il)

where t> is the outside radius of the heat pipe shell. Cotter stated

that heat pipe operation takes place when all temperature gradients are

small except for the radial gradient in the wick and shell. This sim

plifying assumption permitted Cotter to express his energy equation as

QCtf = ^ i [ ^ ~ 1 , (11.12)

where n*. is the liquid specific enthalpy at the liquid-vapor inter

face, hi is the liquid specific enthalpy as averaged over the wick :;;

cross section, and h*a is the heat of vaporization. For the low

radial heat fluxes encountered in heat pipes, Cotter reasoned that the \. ~.

term Q\i - V\'ji)/Ywi could be neglected when compared to unity.

Thus Cotter's resulting relation was

10

Q(& * ^vCy) W (11.13)

Later studies by Chi and Cygnarowicz (5) have confirmed this approxima

tion.

In order to couple the internal phenomena of the heat pipe to

its external environment, Cotter wrote the energy Equations (II.9) and

(II.IO) for an annulus, bounded by y and ^*<A> and radii rv and

i> , and used Equations (l.T.12) and (II..13) to obtain the relation

"T *v T^L ~ ^ r?y-r & T p „ (II. 1*0

Cotter reasoned that, the coupling could be approximated by the simple

thermal conduction result of

T> « Tv -V Q/X'^ . (11.15)

where Q' is the rate of heat transfer externally added or withdrawn

per unit length and XJJLL is the effective thermal conductivity of the

annular rings formed by the wick and solid shell.

Cotter's analysis continued by modeling the external environment

as one of uniform heat addition and uniform heat withdrawal with

QLi) = rfv^YW = X ^ e ;for (II.16)

O £ y 6 Xe. } atsdl (II.IT)

11

QLi) - mvCV)W - -jij^ e v ^or (II.18)

Ae. ^ y * X J (11.19)

where 0e is the total heat input rate at the evaporator. Using

Equations (ll.l6) through (II.19); Cotter evaluated the pressure drops

in "both liquid and vapor regions as determined "by Equations (il.^) and

(II.7) or (II.8). This gave Cotter expressions of the type

L?A - *U*) " PJLCO") o.U \?„ -- P„<o) - Pv(J0 } (11.20)

where t$x and &Pv are "both positive quantities. In order to

relate the pressure distributions of the vapor and the liquid., Cotter

recommended the equilibrium expression of

Pv-P* - *£ , («-21)

where ft* is the characteristic meniscus radius at the liquid-vapor

interface. Cotter's Equation (II.21) neglects the inertia term in the

jump conditions (l^) across the interface. However, typical heat pipes

have low radial heat fluxes, thus permitting the inertia term to he

validly neglected.

Cotter proceeded "by reasoning that at the condenser end of the

heat pipe, n* ; is infinite, i.e. a flat surface separates the two

phases. This approximation allowed Cotter to ohtain from Equation

(11.21) the relation

?*(*) - PJLW (11.22)

12

which when combined with Equation (II.20) indicated

ft«0 -?A(o) -- *£[ L?. + U?y, >, O (11.23)

Cotter then reasoned that the maximum heat flux of the heat pipe would

he limited "by the capillary term 'LO'/rm at the evaporator, and that

this would take place when v - rt where r«. is the characteristic

pore radius at the liquid-vapor interface. Thus Cotter ohtained expres

sions for the maximum heat rate, rewritten as

0. - (, At A l / 'JLr«.cosf

__ Ik. h£l sv e X. g- x.osp

"1

U+ -t rAl? A. ft 7*>J (11.24)

The first, term, containing n<jjt may "be interpreted as a heat

pipe parameter of dimensions power /area which characterizes the working

fluid. The second factor, "n"0r«2'-rJt) m ay be interpreted as the total

cross sectional area of the wick in the axial direction. The third term,

containing e./t is a dimensionless factor which characterizes the

permeability of the wick and the corresponding pressure drop due to

liquid flow:- The bracketed term containing a sm 6 reflects the

effects of pipe inclination in a gravity field. The bracketed term con

taining ><>u/P* reflects the effects of pressure drop in the vapor

region. A

Although the work of Cotter served, as only a first approximation

for analysis of heat pipes, it did provide a basis for other

13

investigations. The work of Kunz., et al. (ijy l8) presented a

simplified approach to Cotter's analytical problem of predicting heat

pipe performance. KUQZ suggested the overall pressure "balance equation

of

(fi>,e-Pv,e) + (P^-Pv,c) +• (f*,c-PJL^ + (fyc - P^e) = O ^ (11.25)

where the subscripts X and;v refer to the liquid and vapor regions

respectively, and the subscripts e and t refer to the evaporator and

condenser sections respectively. For the low velocity vapor region,

Kunz suggested

O v - ^ - ° . (11.26)

Considering the condenser liquid-vapor interface to be f l a t , Kunz used

foe - ?xd ~- ° . (II. 27)

Characterizing the evaporator liquid-vapor interface as a spherical

shape of meniscus radius T^J Kunz recommended

: '0v-•••R.Jiy s ' " - ;^ . (n.28)

Using Darcy's law for liquid flow through porous media and showing the

inertial term to be negligible, Kunz showed

( P ^ - P ^ - K ^ ^ ^ (11.29)

Ik

where K-i is the inverse of specific permeability, kS is the

product of width and thickness for the total cross sectional area of

the wick, m A is the total mass flow rate of liquid, and X-»Ur i s

the effective total length of the heat pipe for computing frictional

pressure drop.

Using an analysis similar to Cotter's for uniform radial mass

flow along the condenser and evaporator, Kunz recommended

m ° "• T i , °»* (11.30)

-L— )

for heat pipes with an adiabatic section, and

- •t'x** .U +4-Ic , (n.31)

A+to r- kx*+k*<- -- xx > (n-32)

for heat pipes without an adiabatic sectioh.

Kunz combined his equations and followed Cotter's reasoning that

the capillary limited maximum heat transfer rate would occur when the

meniscus radius was at its minimum. The resulting expression was

. _ /*rM/^ujt_A Q«*» - \MAA l

/[KJU,J ; (11.33)

where V^u is the minimum meniscus radius. Essentially, Cotter's (8")

and Kunz's (lJP 18) predictions are the same, save vapor effects, with

both requiring wick flow characteristics: for Cotter, tortuosity and

porosity, for Kunz, permeability.

15

Other analytical studies on the general operational .

characteristics of heat pipes include l) sonic limitations due to the

effects of vapor compressibility, Levy (19, 20), 2) time transient per

formance, Kessler (l3) and Biernert (2), and 3) nonconstant fluid prop

erties, Cosgrove, et al. ( T ) • All general theories yield the same

form of prediction, that "being the maximum heat transfer rate is a

function of the physical properties of the working fluid and the

detailed wick geometry.

B. Experimental

Caution must be used in.selecting experimental results to be used

for comparison. Many experimental data have been reported on heat pipe

performance; most operating below the maximum power level and a few at

the maximum.

The experiments of Kunz, et al. (l8) included tests for maximum

heat transfer. Tested was a horizontal water heat pipe of rectangular

cross section, 1.6 inches high by 6.0 inches wide by 2^.0 inches long.

Kunz's wick was formed from sintered nickel fibers shaped to a 6.0 inches

by 2^.0 inches by 0.1 inches sheet. The sheet was bonded to the inner

lower surface of the heat pipe. External heat was electrically supplied

by a resistance heater over a length of 6.2 inches and was removed by

colling water circulation over an adjacent length of 11.6 inches. Max

imum heat transfers were described to occur when high temperatures

appeared in tne evaporator section. Kunz's work also included experi

mentally determined values of Ki j the friction parameter, for various

wicks.

16

The study "by Schwartz (28) also included tests for maximum heat

transfer. His system was a horizontal ammonia heat pipe of cylindrical

cross section, 15-5 inches long "by 0.199 inches inside diameter. The

wick was formed "by cylindrically wrapping two layers of 100 mesh stain

less steel screen. External heat exchange was accomplished "by cir

culating hot water and cold water through jackets around the evaporator

and condenser sections respectively. Each section was 3.0 inches long

with the central 9.5 inches considered adiahatic.

C« Comparison of Theories to Experiments

Comparison of the theories of Cotter and Kunz to the experiments

of Kunz and Schwartz is shown in Figure 2. The agreement is good, with

deviations less than J.0 per cent. The predictions of Cotter and Kunz

coincided for all data points since l) "both experiments used horizontal

heat pipes (no gravity term in Cotter's equation), 2) secondary experi

ments of Kunz on permeability, porosity, and pore size were used to

evaluate the porosity and tortuosity terms of Cotter's equation, and

3) the vapor flow term of Cotter's equation is negligible for "both exper

iments. Kunz's data indicated that condensation took place over a length

of IT.8 inches rather than the design length of 11.6 inches. It- is

suspected that axial conduction was- present in Kunz's thick heat pipe

shell "beneath the condenser wick.

Other experimental maximum heat rates are reported in the litera

ture for capillary limited heat pipes. Comparison is not made due to

l) lack of complete experimental data - Miller and Holm (24), Cosgrove,

et al. (l), and Neal (25), or 2) employment of wick configurations dif

ferent than the porous media assumed "by the theories of Cotter and-S:,'::: :

IT

\<A

Or

\d

\& I O 1

i i—i—r

SYMBOLS :

0 KwNit DKT*

| 1 5 cuvj twvh • o>rk

Nore •. PaeoicrioNS OP COTTER, /\NO KUN t

WJ.e THE sfwve.

J - I I L_L_L l o 3

QN\e*swvt.eo ~ & T U / H R

I I I I I t o *

Figure 2. Comparison of Predicted to Measured Values of Maximum Heat Rates.

18

Kunz - MeSweeney (23)^:grooves>:and Kemme (12)r grooves covered with

screen.

See Appendix V for a calculation of maximum heat transfer rate

•based on Kunz's theory and parameter values tested in this investiga

tion.

2. Parameter Studies

Studying the effects of varying heat pipe parameters consists of

identifying the significant parameters and describing the relations

"between the parameters. The early works of Cotter (8) and Kunz, et al.

(17> 18) served to identify the parameters listed in Table 1. These

parameters include only those applied to heat pipes using wicks formed

of porous media.

A. Fluid Properties

The work of Neal (25) indicated values of the heat pipe parameter

N, where

N = - ± - _ , (II.3M

for various fluids. As seen in Figure ;3> the employment of a particular

working fluid determines the temperature range suitable for heat pipe

operation and the variation of N with temperature.

The parametric analysis "by Parker and Hanson (26) on various

liquid metals included relative values of surface tension and heat of

vaporization, relative values of pressure drop for liquid and vapor

flow.

Table 1. Initially Identified Heat Pipe Parameters

Category Symbol Name Remarks

Operating Characteristics Q* heat r a t e maximum capi l la ry l imit

To operating temperature

average vapor temperature or pipe wall in adiabatic section

Fluid Properties cr surface tension

for liquid contacting solid in presence of vapor

fk

Mji

liquid density

liquid viscosity

liquid flow

Pv vapor density

vapor viscosity

vapor flow

w latent heat of vaporization

energy storage in vapor

#<rVv VJL #A

heat pipe parameter fluid capability

TalDle 1. (Continued)

Category Symbol Name Remarks

Overall t radius of vapor cylindrical heat pipes Geometry space

irv* outer wick radius

rf- outer shell radius

b wick width rectangular heat pipes

5 wick thickness

J£e evaporator length design lengths

/c condenser length

JL. adia"batic length

jj-. total length

<f gravitational significant only when heat pipe -is acceleration inclined in a gravity field

6 . .inclination angle

Overall Geometry

(Continued)

Table 1. (Continued)

Category Symbol Name Remarks

* VY\

Wick Geometry £ pore radius

£ porosity

rim*

*c

t

-Kr

meniscus radius

minimum meniscus radius

contact angle

tortuosity

inverse permeability

"wick structure

liquid-vapor interface

liquid pressure drop

Environmental Coupling

Xf

T*

effective wick thermal conductivity

pipe shell thermal conductivity

evaporator heat source temperature

(Continued)

implied by Cotter (8)

ro H

Category Symbol

Environmental -r-"Coupling

Table 1. (Continued)

Name Remarks

condenser heat implied by Cotter (8) sink temperature

unit conductance.. to heat source

unit conductance to heat sink

ro ro

IO*

If o \fl t-

&

\of*

b 1

I O ' -

lb1- L_

i N I T R O G E N

2 .00 iv-oo

SOOVUIA

WATER

fcOO

23

8oo

T ~ °K

Figure 3. Heat Pipe Fluid Parameter vs. Temperature for Various Fluids.*

*From Neal (25)

2k

The experimental work of Kunz, et al. (l7> 18) included tests

of the same heat pipe operated with fluids of water and freon 1.13. The

water pipe was capable of higher heat transfer rates than the freon

pipe.

B. Geometric Factors

McKinney's (22) work included computer studies for variation of

geometric variables with a water heat pipe. Typical results,, see

Figure h, indicate variation of maximum heat rate with changes in the

overall geometric parameter TV/YV/ and with changes in the wick fric

tion parameter ^ / V%. .

The computer analysis "by Watts (35) included techniques for

optimizing the capillary pore size and thevcorresponding maximum heat

rate.

Edwards, et al. (9) wrote a computer program which included

variations of l) geometrical factors, 2) working fluids, and 3) envi

ronmental factors for a heat pipe operating with the inclusion of inert

gases. Wo graphical results were indicated.

The experimental work of Kunz, et al. (17) included measurements

of the wick friction parameter K, . Their results indicate the valid

ity of assuming Kv to tea constant for a given wick, see Figure 5.

C. Environmental Coupling

Coupling the internal dynamics of heat pipe operation to the

surroundings requires description of the modes and paths of heat exchange.

In his original analysis, Cotter (8) suggested the simple cou

pling model of

IO.OOO

8,ooo

fc,ooo

HEW Pipe.'. w/vrea., HOW^ONTM-

Ty4 = o . 0 3 \ a f t X - I Ft "71 - ?lO «P

a 2 4>ooo

2,ooo

! /K, - 3.25 *iO~ f C

0.0

Figure k. Q v s . r / r for Various l/K.,.-*

*From McKinney (22) ro VJ1

26

\z

\o

8 -

6 -

* ^

Z -

1 " 1 — T T

SYMBOLS :

O t ° 0 N\£SV* SVNTER.EO SCREEN

HI 5 0 tWESH SINTERED SCREEN

o o 0 °"

-ti-a——cr-& £>

X Z 4- fo & to

m / / \ * \o~3 ~ LBtA/west?-

tt \M-

Figure 5. V a r i a t i o n of K wi th Mass Flux R a t e . *

•*From Kunz., e t a l . (17)

27

1) constant temperature or uniform heat flux at the outer

shell radius along the entire evaporator design length,

2) radial conduction through the pipe shell,

3) radial conduction through the wick, i

k) evaporation at the evaporator liquid-vapor interface,

. 5) vapor flow to the condenser section,

6) condensation on the condenser liquid-vapor interface,

7) radial conduction through the wick,

8) radial conduction through the pipe shell, and

9) constant temperature or uniform heat flux at the outer shell

radius along the entire condenser design length. Cotter assumed the

outer shell to be coupled with sinks and sources by appropriate conduc

tances. Studies of Cotter's simple model and its limitations have been

the subjects of many investigations.

The analysis by Lyman and Huang (21) indicated the validity of

assuming simple conduction in the heat pipe wick. This- conclusion was

based on the exclusion of convection since Peclet numbers were small as

calculated from typical heat pipes (lS, 7). Lyman used the suggestions

of Kunz, et al. (18) that the effective wick thermal conductivity be

modeled as isotropic by the parallel conduction relation

Xeff = EX* + (v-e)X5 } (H.35)

where XaH is the effective wick-liquid thermal conductivity, !€. is

the wick porosity, XJI. is the liqiuid thermal conductivity, and X5 is

the thermal conductivity of the wick solid.

28

An experimental study "by Se"ban and A"bhat (3l) included

measurement of effective wick thermal conductivity. Tests were made

during the evaporation of water from two separate screen.wicks, 150 and

325 mesh. Results, see Figure 6, indicated the observed effective con

ductivity to "be "between the values predicted "by simple parallel and

simple series models. The measured values were much closer to the

series predictions. Se"ban did not attempt to explain this trend.

In their similarity investigation, Miller and Holm (2 -) attempted

to overcome the uncertainties of wick conduction, heat transfer through

the pipe shell, and shell "boundary conditions. Their concept consisted

of using a model heat pipe to predict the performance of a geometrically

different prototype heat pipe, "both in an environment of thermal radia

tion. A material preservation analysis was tested in which "both model

and prototype used the same working fluid (water) and wick material (200

mesh nickel screen). This implied equality of thermal conductivities

of the wall and wick, the emittance of the condenser wall surface, and

the permeability of the wick. Using starred quantities to indicate the

ratio of model values to prototype values of that quantity, the sug

gested model equations were

(T,-Ttp; =, _ p y ^ (11.36)

.* . (tfj* Tcp :1 "hff*- > *-U (H.3T)

29

o

SYMBOL N\ooeu N\esv\

o PfcRNxeL 3ZS

0 SERIES 3^5

• PARALLEL 1 5 0

s SERIES I SO

XpRI eovcxeo BTU / v \ R f t ° R

Figure 6. Comparison:of Measured and Predicted Values of Effective Wick Thermal Conductivity*.

*From Seftan and Abhat (3l).

30

//V-crWf/M6

<* - rxcr)Ct.) . c -38)

where Tv and TCp are the temperatures of the vapor and condenser

outside wall respectively, L is the combined length of evaporator and

condenser sections, and hw is the wick cross sectional area. Experi

mental results indicated the prototype "behavior could "be predicted from

model performance to within 10 F over a temperature range of 1 0 -

330° F.

In a recent paper, Sun and Tien (3^) constructed a simple con

duction model for theoretical steady, state heat pipe performance. By-

considering the interaction of axial heat conduction in the shell and

wick, the effective lengths of condensation and evaporation were shown

to "be larger than the design lengths. For a given pipe geometry,

these effective lengths and the corresponding overall conductance of

the heat pipe were shown to depend significantly on .the two dimension-

less parameters M and BA } where

M - — 1 - K L A J L L , «4 (H.39) (~ — —I '

I H ^

M has physical interpretations of the ratio of wick to shell wall con

ductance times the ratio of shell surface area to shell cross sectional

31

area. B;. the Biot number, has a physical interpretation of the ratio

of wick radial temperature drop to the temperature difference "between

the outer shell wall and the surroundings. Sun recommended that axial

conduction "be neglected for values of M greater than 100 and Bi less

than unity. See Appendix V for a calculation using parameter values

tested in this investigation. '

Fox, et al. (lo) experimentally operated heat pipes with con

figurations similar to that proposed in this investigation "but with dif

ferent values of the geometric parameters. His system was a horizontal

heat pipe of cylindrical cross section, 2k inches long "by 1.50 inches

outside diameter with 0.028 inch shell thickness. Water was the

working fluid. The wicks were formed "by cylindrically wrapping mesh

screen. Tested were a 100 mesh wick consisting solely of layers of 100

mesh screen and a dual wick consisting of l6 mesh screen layers covered

with a single layer of 100 mesh screen. Operating temperatures (incom

pletely reported) were approximately 200 F with power levels from 400

to 1250 watts.

Schwartz (29) also tested a heat pipe with configurations similar

to that proposed in this investigation shell and wick were made of

stainless steel with water as the working fluid. Geometries were the

shell, 15.5 inches long "by 0.199 inch inside radius with 0.020 inch

thickness, and wick, two tightly wrapped cylindrical layers of 100 mesh

screen. Operating data were collected over the ranges 79 "to 153 F and

6.3 to 58 watts.

32

CHAPTER III

GEHERAL THEORY

1. Objective

The purpose of this analysis is to study the general relations

which describe the performance of a heat pipe. Results of this study

will l) provide a list of governing field and constitutive equations,

2) provide a list of potentially important dimensionless groups, and

3) provide an analytical "basis for future studies.

2. Overall Concepts

The physical model of the heat pipe, see Figure J,, consists of

an annular region of porous material "bound "by a circular cylinder at

r= Vw and flat planes at each end. A fluid is distributed within

the pipe. The pipe is divided into the following axial sections: l) a

section surrounded by a heat source, the evaporator section, 2) a

section surrounded by insulation, the adiabatic section, and 3) a

section surrounded by a heat sink, the condenser section. The pipe is

divided into the following i adial regions: l) a vapor region,

r < »W*F*ce ; 2) a liquid-wick region, r1NT(illFftce < r <c r« t

and 3) an external region, *" > *™ . The working fluid is modeled as

being distributed in the liquid phase solely within the liquid-wick

region and the vapor phase solely within the vapor region.

The general formulation is made by separate studies of each region

of fluid with appropriate boundary conditions and using reference param

eters to obtain dimensionless groups.

33

-*• VftPOR

UQU\D

JU BYAPORM-OR PkOlNBKTlC COK0e.NStR

Figure 7. General Theory Heat Pipe Schematic.

3

3. Reference Parameters

Table 2 indicates the nondimensional parameters as defined in

terms of the dimensional parameter and reference parameters. For

material properties, the general relation is

. . . , dimensional property , ___ _ >. nondimensional property.= ———•—— — , -v (III.l)

J- ^ reference property . '

The nondimensional parameter is indicated by the property symbol with

a bar drawn over it. The dimensional property is interpreted as the

actual value of the property at its local spatial location and tempera

ture. The reference parameter-is the va,lue of the dimensional property

at temperature T<, ) where T,e is defined as the heat pipe operating

temperature, interpreted as the average vapor temperature or wall tem

perature in the adiabatic section. Liquid and vapor reference proper

ties are saturation values evaluated at To. Exceptions to this system

are l) vapor enthalpy,, and 2) liquid and vapor pressures where specified

reference parameters are used.

Geometrical parameters are nondimensionalized by their indicated

reference parameters.

Velocity terms are nondimensionalized by the reference velocities

of

Qe U * = ftCvrrvJl^W , ^ ( I I I . 2 )

u* "- 'fifcrnA^ ' (HI-3)

Tahle 2. General Theory Reference Relations

Category Name Reference Relation

Temperature temperature _ T

Vapor density

pressure ftUv

viscosity A v = Mv

thermal conductivity x - -*-NV

enthalpy Y\

V\y - f T fv

Liquid density e

pressure P* = /W

viscos i ty ^JL ~ Mji

(Continued)

Table 2. (Continued)

Category Name Reference Relation

L i q u i d t h e i ^ n a l c o n d u c t i v i t y XjL -" x x*

surface t ens ion cr -(T

(TrtTo

entha lpy K- CpATc

Wick P r o p e r t i e s

w i c k - s o l i d t h e r m a 1 c o n d u c t i v i t y %

_x_ Xs

wick effective thermal conductivity

X€ff X«K X«« at To

wick friction factor (inverse permeability)

K

Vapor Trc Velocities radial velocity Uv =•

Ul Uv

angular velocity - J£_ "^v/ = ULv

(Continued)

Table 2. (Continued)

Category Name Reference Relation

Vapor Velocities

axial velocity UJ

uJv = 7.uu(JL/r

Liquid Velocities radial velocity — ^

UA= "Si

angular velocity _ -vr

axial velocity uu

<% = L^f-^-NujL e0 rv Jjy-U

Bulk Geometry radial coordinate

r

angular coordinate e = -^TF

axial coordinate a, = X-

Detail Geometry radial coordinate rs = rc

(Continued)

Ta'ble 2. (Continued)

Category Name Reference Relation

Detail Geometry radial coordinate i -- •-*.

radial coordinate Is 2k. Tc

angular coordinate 6,

axial coordinate V« li

surface coordinate ?

surface coordinate <ft n

wetting angle <t> '- *

porosi ty

(Continued)

39

Table 2. (Continued)

Category Name Reference Relation

Environmental _ • Parameters heat transfer rate Q =. —r—

^ Qe

unit conductance \j •=. Uc

Environmental references parameters are l) Tc, the average

temperature of the heat sink, 2) U C ; the average unit conductance

"between the heat sink and the outer surface of the wick;, iandr 3) Qe, the

total -power .delivered to the evaporator section from the heat source.

h. Vapor Region

The vapor region is studied "by examining its spatial "boundaries,:

assumptions, field equations, and constitutive equations. Dimensionless

groups are obtained.

A. Spatial Boundaries,

The spatial "boundaries, see Figure 8, are

o ^ y < U +JL -vie , '(III. M

o ^ r *. rv +- *)(d,fr) t «*A (III.5)

o <. e ^ 2-TT . (III.6)

ko

a c ne

ne -*- V\L

^

M

y

A*

Yw

Figure 8. Genersil Theory S p a t i a l Boundaries .

hi

B. Assumptions

The "basic assumptions are:

1) single species,

2) no hody forces,

3) no heat generation,

k) negligible heat exchange "by radiation,

5) no phase changes occur within the spatial "boundaries,

6) steady.state,

7) laminar flow, and

8) negligible viscous dissipation of heat.

C. Field Equations in Vector Form

The field equations in vector form are

equation of continuity

V-(pv) - o } (III.7)

equation of motion

v-(pvv) •.- - V P - v-r } awi

equation of energy

v-(pvK) - - v«^ -v v- vP

(III.8)

(HI. 9)

D. Constitutive Equations in General Form

The constitutive equations in general form are

rheological equation

r = T(>> )>V!£) ) *«• (III. 10)

k2

j^l,Z,3 a*A k= l,-Z-,3 , (III.11)

equations of viscosity

u -- /*("0 } <MU ( I I I . 12)

X - \ (T) , *We ( I I I . 13)

A = "* ~^M. for too«\».torMC <jaS£S ( I I I . 1 4 )

heat conduction equation

% ~- - X V T t (III. 15)

equation of thermal conductivity

X = XCT) ; (III. 16)

thermal equation of state

P' />(^T) J anA (III.17)

caloric equation of state

K - MP,T) . (III. 18)

E. Dimensionless Groups

Writing Equations (ill.h) through (lll.l8) in dimensionless form

with the reference parameters, see Appendix A, results in the following

seven dimensionless groups for the vapor region

ii ii. H !L fri^" Mv/C?,/ uft

3

5. Liquid Region

The liquid region is studied "by examining its spatial "boundaries,

assumptions, field equations, and constitutive equations. Dimension-

less groups are obtained.

A. Spatial Boundaries

The spatial "boundaries, see Figure 8, are

(111.19)

(111.20)

(111.21)

O £ y ^ JU + io. + JL ;

rv + ^(e^i •< r <. *# ( **l

o ^ e <. ^ T

B. Assumptions

The "basic assumptions are

1) single species,

2) no "body forces,

3) no heat generation,

h) negligible heat exchange "by radiation,

5) no phase changes occur within.the spatial "boundaries,

6) steady state,

7) constant density,

8) negligible viscous dissipation of heat,sand

9) Darcy's law for flow in porous media applies.

C. Field Equationsc:in Vector Form

The field equations in vector form are

equation of continuity '

V-(P^) = ° , (III.22)

t

equation of motion

kk

V ' ( p V v ) = - V P - 7 ' T } cxrxi ( I I I . 2 3 )

equation of energy

' • ( f V W) - - ^ ' % + V • V P ( I I I . 2 4 )

D. Constitutive'Equations in General Form

The constitutive equations in general form are

rheological equation

V»r = - KM £V (III.25)

equation of viscosity

M. = M (T) (III.26)

equation of wick friction parameter

K^ K( (III.27)

equation of porosi ty

& ~ ^- ky^") ^ ; ^ S ; "«. j * M c k co* \ f vovro.t \OY\) f o r ( I I I . 2 8 )

ev ipr\ft.SS •floyx yoAe M"> >. JUrec-tv

*• ~ to"t»-l cross sect»<»^*.l are*, m *X" <k\rec/t\orv (HI.29)

heat conduction equation

% "- -X*T (III.30)

^5

equation of liquid thermal conductivity

X* = X*(r) ^ (III.31)

equation of wick-solid thermal conductivity

x* = x 5 c 0 , ( in.32)

equation of effective wick thermal conductivity

X«*f = X ^ ^ X A C T ^ X S C T ^ Y ^ , r., **,, v%) ; (III.33)

thermal equation of state

P - corNstarxt ; an<H ( i l l . 3 ^ )

calor ic equation of s ta te

h « h(P,T) # (III.35)

E. Dimensionless Groups

Writing Equations (III.19) through (ill.35) in nondimensional

form, see Appendix A, results in the following fourteen dimensionless

groups for the liquid region

JU SU JQL : J[L IL V«_ ' VA : K**_ A<. , U , ^ ; rv , J L , X A , x . « /*"«• J K|V"C* J 6 o ; n ;

ftlA)Lt-c Aft C f l _ iCfc

^JL ) XjL , C f j l T ^ .

he

6. Liquid-Vapor Interface

The liquid-vapor interface, that surface separating the liquid

and vapor regions, is studied "by examining its spatial location, assump

tions, "boundary conditions, and constitutive equations. Dimensionless

groups are obtained.

A. Spatial Location

The spatial location, see Figure 8, is

o'^ y ± JU + ^ + 1 ; (111..36)

r =: n, + y\ie^ i *U (ill.37)

o ^ e T . (ill.38)

B. Assumptions

The "basic assumptions are

1.) steady state,

2), negligible heat exchange "by radiation,

3) negligible effects of dilatational and surface shear

viscosities,

k) constant liquid density,

5) continuum flow in regions surrounding the surface, and

6) negligible viscous dissipation of heat.

C• Boundary Conditions in Vector Form

The boundary conditions in vector form are

equation of continuity

(Pv^V^x + C P A V ^ - o , (III.39)

4?

no s l ip conditions

(vv-v^.t, -- o ; (in.4o)

(vfg - V A y t x = O , ( I I I . 41)

(VjL -"V>Vt% "- O ; a*A ( I I I . 42)

(VA-v^-t^ ^ o , ( in .43 )

equation of motion

(fv w)-^v v + (p^v,). h,Vj[ = -P vfu- (in.H)

-P<nv - f ; . ^ -Tx.«, - 3 -v.o- ?

equation of energy

[ fVvv ( \>v 4 ) + V -Tv-v^n. +•' [ ft \/, (^ +£) * (m.45)

D. Constitutive Equations in General Form

The constitutive equations in general form are

Theological equation

=. = / **-

T - T l^ ) , J ; fov (in.46)

H =- \ -L , S *wJ> V- \i,3 ; (III.4T)

equations of viscosity

kS

Av = A*Lr) , a*4 ( I I I . 48)

MA = yU^CO } ( I I I .49)

equation of heat conduction

% = - X 7 T , ( I I I .50)

equations of thermal conductivity

XJL = X.*(TN> , , ( I I I . 51 )

X„ .= XvC-ry;,^Jl ( H I . 5 2 )

Xs = X 5 ( T ) , ( I I I . 53)

thermal equations of s ta te

A = PVO>,T) ja*Ji ( H I . 5 4 )

ft = frO\T) ) ( I I I .55)

caloric equations of state

Vv, = V\„C?,T) , **A (III.56)

W = W C ^ T ) ; (III.57)

equation of latent heat of vaporization

V\\ijL = \\\> ~ rVjt oA. s«."twv(xtiors ( i l l . 5 8 )

equation of surface tension

r = °^T) ; (III. 59)

k9

unit normal vectors

^ - - ^ - T\(**rface ^oweir^) ^ (III.60)

unit tangent vectors

_» _ i

"-M = "*-• \ (^sw.rfa.ce ^ e o t ^ e t r i j ) } o.n£ (lll.6l)

t - t = t-t.^SttcfoLce, ^eo«\etr«j) (ill.62)

radius of curvature

Rm - RTA (•s^rf&te <jeow\etr«j) } (ill.63)

surface gradient

V 3 - • o p e r a t o r basedl OY\ Sur face ^ e o m e t r M ( i l l . 6k)

surface divergence

V5» "*" operator based ow surface e>eoti\e-tTu j a J[ ( i l l . 65)

surface coordinates

f = •?(*<•>*«,***,.4) , <^J (III.66)

ft '- f^^S**, <j>} . (III.67)

E. Dimensionless Groups

Writing Equations (ill.36) through (ill.67) in dimensionless form

with the reference parameters, see Appendix A, results in the following

twenty dimensionless groups for the liquid-vapor interface

50

JU JL Y\^ rv Y£L TV. j ^ _ g> tv &<_ f^rcUt *«. , JU , w , "XI , > c j rv , p„ , jki j ~~AT J

fta*![c a V ^ c f i ^*gjL _i>L- u*.._ X5_

7*" * ^ *

7. Environmental Boundary Conditions

The environmental "boundary of the heat pipe., that surface

separating the contained working fluid and its surroundings, is studied

"by examining its spatial location, assumptions, "boundary conditions,

and constitutive equations. Dimensionless groups are obtained. The

"boundary is subdivided into the following five sections: evaporator,

adiahatic, condenser, evaporator end,; and condenser end. These sections

are distinguished from each other "by their respective spatial locations

and their respective constitutive equations of environmental heat

transfer.

A. Spatial Location

The spatial location of the heat pipe "boundary sections, see

Figure 8, are as follows

evaporator

r= rvu (III. 68)

O ^ V i i e ,OJ4 (III.69)

51

o ^ e ^ ' • ( in .TO)

adiabatic

r-- w, , ( I I I . 7 1 )

JU <• ^ <• X* +• . JU j c U ( i n . 7 2 )

o ^ e <r -2.K ; (in.73)

condenser

r = r w > ( i l l . 7*0

J U + 4 * ^ y • < • X e v JU + I c ., *n«A ( I I I . T 5 )

° ^ 0 6 -LTT f ( I I I . 7 6 )

evaporator end

• ° ^ ^ * r * ; ( I I I . 7 7 ) V

> = o , **<* ( I I I . 7 8 )

o . © . -LIT/ o.«Jl ( i l l . 7 9 )

condenser end

o ^ r < r w , ( i l l . S o )

^ = A c + ^ + I c , <xn<A ( I I I . 8 1 )

O 6 0 <r <LTT . ( 1 1 1 . 8 2 )

52

B. Assumptions

The basic assumptions which apply to each subdivision of the

boundary are

1) steady state,

2) continuum flow on fluid sides of the boundaries,

3) the environment contains no working fluid,

k) the boundary is motionless,

5) negligible heat exchange by radiation, and

6) negligible viscous dissipation of heat.

C. Boundary Conditions in Vector Form

The boundary conditions which apply to each subdivision are

equation of continuity

- O (III.83)

no slip conditions

(pv).t, -- o }^i (III.8k)

(HI. 85)

equation of energy

[PV(Y\*£) +%\-ti A- {$*[•% - O ( I I I .86)

D. Constitutive Equations in General Form

The constitutive equations which apply to each subdivision are

equation of heat conduction

% - -X VT (III.87)

equation of liquid thermal conductivity

X* = XA (J) (III.88)

equation of wick-solid thermal conductivity

X, = Y.9(r) } (III.89)

equation of wick effective thermal conductivity

X«H = Xe f (XJI(X^XS(T), r*} r„,. r«.} Y\,S 3 vC) (III.90)

liquid thermal equation >:of state

P. •= C<9V\st.<X.Y\'t

liquid caloric equation of state

W = \\AC?,T) J

unit normal vectors

*> = -** - * ( Y ,9) ;

unit tangent vectors

ti = "t-i (f, \ j &) , <x*4

(in. 91)

(III.92)

(III.93)

(III. 9*0

(III.95)

The remaining constitutive equations which apply only to the respective

subdivisions are

evaporator section

5k

external heat flux vector

STLVo^H'^ = ®* >

adiabatic section

external heat flux vector

(III.96)

JQS'X.L.-C....

-V = Q ; (III. 97)

condenser section

external heat flux vector

fix ( JU+JL+ ic vuff f JU+-JL+ic

k 3i,d5«'&--*W*• -- J. i i . ^ T l w - ^ ' , ) _ T J r- **Je > ( n i-9 8 )

where Uc and Tc are reference parameters, see Chapter III, Section 3>

and

evaporator and condenser end sections

external heat flux vector

%* = ° , (111.99)

equation' of vapor thermal conductivity

X,- X„(T} , (III. 100)

vapor thermal equation of state

ft = M P J T ) , **<* (III.101)

vapor calorie equation of state

55

Kv = k,(?>T>) - (in. 102)

E. Dimensignless Groups

Writing Equations (ill.68) through (ill.102) in dimensionless

form with the reference parameters and substitution of the constitutive

equations, see Appendix A, results in the following eleven dimension-

less groups for the environmental boundary conditions

i« lo_ Yy_ T«_ T^ TW Ys_ X«ff Ac ; Ac , U , rw, r„ , ro , XJL , ~yT , n

8. Heat Pipe, Operating Temperature

Temperature, the explicit parameter appearing in various consti

tutive equations (e.g. equation of thermal conduction), provides the

dimensionless group

T0r>ir,«>

Tc

where Tc is the condenser sink reference parameter. In accordance

ith the reference parameters, see Chapter III, Section 3.? this group

is interpreted as

Tc ~ ; ; . "

w

where Ti> is the heat pipe operating temperature,

9. Summary of Dimensionless Groups

The collection of 2h dimensionless groups obtained from the

general theory is shown in Ta"ble 3. For groups containing velocity,

the defining reference parameters of Equations (ill.2) and (ill.3)

have "been substituted, hence velocity does not appear explicitly. All

symbols refer to the reference parameters, see Chapter III, Section 3«

Details are indicated in Appendix A.

Ta"ble 3. Dimensionless Groups of General Theory

Category

Material Properties

Syrnbol

Jk ft

Remarks

liquid density vapor density

A/Cp» vapor Prandtl number

MjiCfSL

" X A l i q u i d P r a n d t l ntimber

h-Xje

•wick-solid c o n d u c t i v i t y ?;1;i|i4uid c o n d u c t i v i t y .

Cpy

C-fil

vapor specific heat liquid specific heat

w cpJiTc

Kutateladze nuniber

X«H

X* wick effective conductivity . -;,, liquid ;• conductivity,.' .

Flow Parameters -IT v^rv>u.vj vapor a x i a l Reynolds." number-

c o n t i n u e d )

57

Table 3. (Continued)

Category 'symbol Remarks

Flow Parameters

. / Qe " -*— 1 n—7\T \ vapor Eckert number

L£—-[ %—X\ V Weber number o- VuKniJtcftivi)

\r3T-)CK'*). Darcy number

Q< irrcVvn. AJL

liquid pore Reynolds 'inumber;

Geometry r*

vapor radius wick outer radius

vapor radius condenser length

ii. JU

evaporator length condenser length

X adiabatic length condenser length

wick pore radius vapor radius

fes 'YV."

wick-solid radius wi ck porer radlu s

t\ number, of wick screen layers.

(Continued)

58

Table 3- (Continued)

Category Symbol Remarks

Geometry £ fluid wetting angle

W evaporator meniscus radius

wick pore radius

wick porosity

Environment II Tc

heat pipe operating temperature eocondenser sink temperature

Ucic

XJL

liquid thermal resistance condenser external thermal resistance

59

CHAPTER IV

EXPERIMENTATION

1. Objective

The purpose of this experimentation is to provide data from an

operating heat pipe whichwl11 allow correlation of the parameters and

further understanding of heat pipe operation. The results of these

experiments will yield measured values of temperatures and heat trans

fer rates for a horizontal heat pipe of fixed geometry with a working

fluid of water and methanol, and provide raw data which can "be used to

evaluate the parameters required for correlation.

2. Equipment

The equipment is comprised of the heat pipe with its attached

instrumentation, see Figures 9> 10, and 11, and the auxiliary equipment

and instrumentation, see Figure 12.

A. Heat Pipe

The heat pipe shell, see Figure 9> consists of a 3/4 inch diam

eter "by l8.0 inch long tube of 304 stainless steel. The heat pipe wick

is composed of two layers of 3l6 stainless steel screen, 100 mesh and

0.0045 inch wire diameter, mounted concentrically on the inside wall

of the pipe shell. See Table 4 for a listing of geometric parameters.

The condenser section is constructed from a 5-0 inch long "by

1 inch diameter schedule 40 pipe of 304 stainless steel. The pipe is

solder mounted concentric to the heat pipe shell and sealed at each end

EVAPORATOR

TRANSDUCER

NOT TO SCALE

Figure 9. Exper imental Heat Pipe Dimensions.

6l

.2.WM**V**v'e',fc £.«>&" O.O- * \ - 3 ^ ' VO. * O.ZS"\NVOt WWC

SCM.G : I'Vl = l"

Figure 10. Experimental Heat Pipe.Condenser Header,

. 4-M-3(9

Hj, Fp^Xc

U '-fe7'

< — h

L__UI

M-5

3 7

1C

zs

\ . te?"

tfe 127 Z8

^

39

*

3? 33 A x_L. J * « U

M-(o

m

3.H-i" ~i

o.fc o.a

lo<l

o.8"

t i l

Q.84-

H3 US . Q Q O . Q O O . O O O . O O O 7

TA 3 © i.feS"

Z7T

If 15 \lo \1 18 ft. K 3 t

ry 3&

O THEP- TH£W«OCOUPLES *• | LOW CAUB. FROM ICE

2.3 H»CH CftUft. F ROM POT. Z4- R6V.AV SKIP 4-7 ROOM TEMPERATURE 4-8 RELA* SK\P

J.d 2 . 0 t . O

I Z

"i.o

13

V.7t

*Q O Q*f l fl O'Q O O ' O O O'O »ofe tog no it-2. t t f

o.S' o.8" 0.8 oS o.z8

4-o 4-1

L / F o v u UNfc OF IHSUV-/VTVON

tOl UtGH C A U S . ' T O RECORDER I OS It X&CTlOM P«.e-HEM"

Figure 11. Experimental Heat Pipe Instrument Location. ON

ro

m z

rz

<<w-

• «

'[

A ^2^

it, >°6

JL

39'

a. SE 3 37 -«2\

H ijh'—— r i ^ + Xf u-t-J^- --'»*,

/ < I j F-f- - - T - - S < f f

/ Tv1 ! ' ! 3<0 JI I u o*

hi. "C

"37

OOH CSL «3 14-

U-o

<M 3X 33 34-

3 5

TT 4-^

Q 15

to

17

r—/l / A N O T E :

Sfcfe TI*,Bt.e. (a Foe £S>vJvPN\eKT »-\ST

Figure 12. Experimental Auxiliary Equipment Schematic

Table 4. Geometric Parameters of Experimental Heat Pipe.

Value Name Symbol inches (u.o.n.)

evaporator length He. 3-36

adiabatic length JU 9.6k

condenser length X, 5.00

inside wick radius w 0.298

outside wick radius n- 0.326

wick pore radius rt 0.00275

wick-solid radius Vws 0.00225

number of screen layers ri 2 layers

65

by header chambers. Each header is composed of stainless steel plates

separated by a l/k inch wide spacer ring. Openings for cooling water

flow consist of a single port in one of the plates and multiple holes

in the condenser pipe between the plates, see Figure 10. The condenser

end of the heat pipe is sealed by an 0-ring between the end header plate

and a flange.

The evaporator section is formed by an electric resistance heat

ing coil wrapped helically around a 3«3^ inch long section of the heat

pipe shell. The coil is covered by a sheath of brass foil. The

evaporator end of the heat pipe is sealed by a welded plate of 30 - stain

less steel.

The adiabatic section is covered with fiberglas insulation. The

entire heat pipe (condenser., adiabatic., and evaporator) is covered with

fiberglas insulation to a diameter of 5«5 inches and is covered with

aluminum foil.

Heat pipe instrumentation., that instrumentation attached directly

to thehheat pipe test section, consists of two pressure transducers and

various thermocouples, see Figure 11. The condenser pressure transducer

is mounted to a copper tube positioned on the-heat pipe shell in the

adiabatic section adjacent to the condenser header. The evaporator

pressure transducer is mounted to a copper tube positioned on the heat

pipe shell in the adiabatic section and adjacent to the end of the heat

ing coil.

Thermocouples are mounted on the heat pipe in seven different

areas, see Figure 11 and Table 5« Wick thermocouples are spot welded

between the two layers of screen. These wires extend out of the heat

Table J?. Thermocouple Materials

Category Junction Wire Size Insulation

Condenser Copper-Wall . Constantan 36 ga enamel

Adiabatic Wall

Copper-Con stan tan 24 ga fiberglas

Evaporator Wall

. Chromel-Alumel 30 ga fiberglas

Wick Copper-Con stan tan 36 ga enamel

Cooling Water.

Copper-Con stan tan 30 ga nylon'//

Vapor Probe

Copper-Con stan tan 36 ga

Ceramic in S. S. Sheath

Insulation Copper-Con stan tan 2k ga fiberglas

67

pipe through an epoxy sealed hole in the condenser flange. Condenser

wall thermocouples are mounted in grooves in the outer shell wall. The

grooves are sealed with a epoxy of high thermal conductivity. Adiabatic

shell wall thermocouples are spot welded to the outer shell wall. Evapo

rator shell wall thermocouples are spot welded to the outer shell wall

between the coils of the heating element. These wires extend out through

the "brass foil. Vapor thermocouple probes are positioned at the center

of the evaporator and condenser sections. The probe sheaths are attached

to end plates by silver solder. Insulation thermocouples are mounted on

the underside of the aluminum foil. Cooling water thermocouples are

epp-xy sealed within the tubing attached to the ports of each condenser

header.

B. Auxiliary Equipment

The auxiliary equipment is made up of systems for l) vacuum,

2) condenser cooling water, 3) evaporator electric power, k) pressure

transducer power supply, 5) working fluid injection, and 6) emf measure

ment, see Figure 12 and Table 6.

The vacuum system consists of the vacuum pump, connecting copper

tubing, a valve attached to the condenser end flange, and a top line of

copper tubing to the Alphatron pressure sensor.

The condenser cooling water system is composed of a constant head

tank supplied with building water, precooling chamber, heat pipe inlet

control valve, water collection beaker, and connecting tubing.

The evaporator electric power system is comprised of the alter

nating current voltage regulator, volt meter, ammeter, and watt meter.

68

Ta'ble 6. Equipment List

Equipment numbers refer to Figure 13,

Item No

1

Name

Recorder

Alphatron Pressure Gage

Potentiometer

Pre-chiller

Ice Reference

Thermocouple Panel

Thermocouple Panel

8 Relays

9 Valve

10 Valve

11 Alphatron Sensor

12 Switch

Description

Honeywell Model SY153-89-(C)-II-IH-16 2k channel

NRC Equipment Corp., Alphatron Type 530

Leeds & Northrup No. 8686

ETDCO water cooler, D 89031

Dewar Flask

For thermocouple extension wires from relays

For thermocouple extension wires to recorder

-8 "bank relays, Southern Bell

Nupro Type B-^H

Nupro Type B-^H

For Alphatron Type 530

For 10 Potentiometer thermocouples

13 Switch For 10 Potentiometer thermocouples

Ik Switch From power source to relays

15 Vacuum Pump Cenco-HYVAC ±k, No. 91705

16 Stop Watch Compass, 1 jewel

17 Collection Water Beaker Pyrex No. 30^2

' 1 8 ' •-"'•* Support :" Bar for heat pipe support

(Continued)

69

Table- 6. (Continued)

Item Wo Name

19 Support

20 Condenser Pressure Transducer

•21 Evaporator Pressure Transducer

Description

22, Injection Burette

23 Constant Head.-Tank

24 Voltmeter

25 Voltage Regulator

26 Ammeter

27 Wattmeter

28 Voltage Regulator Power Supply

29 Voltage Regulator

30 - Power Panel

31 Rectifier

32 Switch

33 Switch

3h Switch

35 Wattmeter

36 Heat Pipe

Bar for heat pipe support

Statham Model PA769-50

Statham Model PM732TC + 25-350

LKMAX Type TD, with nichrome ribbon

Shown with sight glass

Simpson, 0-150 AC

Powerstat, 0-115V AC, for evaporator coils

Simpson, 0-25 AC

Triplet, 0-750 W (not used in tests)

Lambda Model LCS-2-01, 0-6.5V DC, for excitation to transducers

Powerstat, 0-115V AC, for relay

For connections to building lines

0-30V DC output

To relay bank

To injection burette coil

To evaporator heating coils

Weston Model 310

Shown with evaporator coil of Chromalox- Type- TSSM^O, • 1120- w.

(Continued)

Table 6. (Continued)

Item No Name

37 Valve

38 Valve

39 I n s u l a t i o n

^0 Vacuum Tubing

lj-1 Cooling Water Tubing

k2 Table

^3 Test Board

hk Tank Support

Description

Nupro Type B-^H

Nupro Type B-4H

Fiberglas with Aluminum foil cover

Copper, to vacuum pump

Copper., to collection beaker

Wood

Plywood with angle frame

Steel angle

71

The transducer power supply system is a single direct current

power supply for both transducers.

The working fluid injection system utilizes a 50 ml. burette,

resistance heating ribbon wrapped around the burette, and an injection

valve joining the burette to the heat pipe shell. A l/l6 inch diameter

hole in the pipe shell permits the flow of injection fluid.

The thermocouple recording system consists of a 2k channel

recorder, 48 channel relay bank with power supply, and an ice reference

flask. Direct emf measurement is accomplished with a potentiometer,

potentiometer switches, and the same ice reference flask.

,3• Procedure

The operational procedures of this study are composed of various

stages of construction and testing.

A. Heat Pipe Construction

Construction consists of

1) assembling the parts of the heat pipe shell, cooling water

jacket, and evaporator heating coil,

2) attaching the shell wall and vapor thermocouples to the heat-

pipe,

3) attaching the pressure transducers,

h) spot welding thermocouples to the wick, forming the wick

around a core, spot welding the wick along its seam into its final form,

cleaning the wick with acetone, and inserting it into the pipe shell,

5) sealing the condenser end with 0-ring and flange, and sealing

the wick thermocouple port with high vacuum epoxy, and

72'i

6) insulating the entire heat pipe,, attaching insulation

thermocouples, and foil attachment.

B. Preliminary Testing

Preliminary testing is comprised of instrument calibration and

vacuum leak testing.

Each transducer is calibrated "by recording the emf signal,

measured with potentiometer, and the column height of a mercury monom-

eter. These calibrations are made prior to transducer attachment to

the heat pipe.

Thermocouple calibration consists of recording the emf signal,

measured with potentiometer, and indication "by a standardized thermom

eter, "both in a constant temperature "bath. For thermocouples with fiher-

glas insulation, a sample thermocouple is calibrated,... All other thermo

couples are individually calibrated. These calibrations are made prior

to thermocouple mounting onto the heat pipe.

Vacuum leak testing is composed of drawing a vacuum on the heat

pipe interior, closing valves to the surroundings, and recording the

internal pressure over a time period, see Appendix B. The Alphatron

vacuum gage is used for pressure indication. These tests are repeated

until all leak points are sealed either with solder or epoxy. The heat

pipe is considered sealed when the leak rate is sufficiently low for

only negligible amounts of room air to enter the pipe over the time

period required for data taking. These tests are conducted prior to

working fluid injection.

C. Start up

Operational start up is "begun with vacuum pumping over a 2k hour

73

period. The working fluid is then pre-heated with the resistance

ribbon on the burette., Heating takes place for 30 minutes with no

boiling. Injection is accomplished by opening the injection valve and

permitting liquid to be drawn into the evacuated heat pipe. The amount

of fluid injected is that amount estimated to saturate the wick with

liquid and fill the vapor region with vapor, see Appendix C. External

heat transfer is initiated by first flowing the pre-chilled cooling

water through the condenser jacket and then raising the voltage across

the evaporator jeating element until the desired power level is reached.

D. Measurements

Data acquisition consists or recording.four types of steady state

information l) condenser conditions, 2) evaporator condition, 3) general

emf data, and k) room conditions. The system is considered to be at

steady state when no observable changes in any data occurred during the

time period (nominal 15 minutes) required for data taking.

The volume flow rate of cooling water is measured by collecting

the water in a graduated beaker over a measured time period. Other

condenser conditions are the potentiometer readings for inlet tempera

ture and temperature difference between inlet and exit.

The power (wattage) delivered to the evaporator heating coil is

recorded.

General temperature and pressure data are comprised of

l) potentiometer readings of transducer emfs, 2) potentiometer read

ings of evaporator shell wall thermocouples, and 3) recorder printings

of emf signals through the relays for both banks of 2k thermocouples.

Room temperature and barometric pressure are recorded.

n:

E. Change of Operating Point

The heat pipe operating point, characterized during testing by

adiabatic wall temperature and coil power, is changed in one of three

ways. These methods are l) change the power level delivered to the

evaporator heating coil, 2) change the water flow rate through the con

denser cooling jacket, or 3) change the cooling water inlet temperature

by alteration of the percentage of pre-chilled water flowing into the

cooling jacket.- Once the change is initiated, the system is permitted

to reach steady state, then data taking is repeated.

F. Shut Down

When a series of tests is completed, the evaporator power is

turned off. After the evaporator vapor thermocouple reaches room tem

perature, the condenser cooling water flow is terminated.

G. Terminal Testing

Following heat pipe shut down all thermocouples reach room tem

perature (nominal 12 hours). The transducer pressures are then measured

and recorded. A negligible difference between transducer pressure and

interior saturation pressure indicates no gases leaked into the heat

pipe during operation.

h. Results

Experimental results are comprised of three parts l) direct data,

those data recorded during heat pipe operation, 2) data reduction,

application of calibration and correction factors to direct data, and

3) listing of the reduced data.

75

A. Direct Data

Direct operational data consists of l) power delivered to heating

coil, 2) volume of cooling water collected, 3) time lapse of collection,

see Appendix D, and k) emf data, see Appendix E.

B. Data Reduction

Data reduction is divided into the three parts of pressure and

temperature reduction, evaluation of the heat pipe evaporator heat trans

fer rates, and evaluation of condenser environment conditions.

. Data from pressure transducers and thermocouples are reduced with

calibration curves, see Appendices F, G, and H. The heat pipe heat

transfer rate is evaluated by estimating the heat loss rate to the

surroundings, see Appendix I.

The condenser sink temperature is evaluated from cooling water

temperatures, see Appendix J. Evaluation of the unit conductance between

the sink and the outer wick wall is accomplished by considering the tem

peratures, heat rates, and heat flow areas, see Appendices K and Q.

A discussion of the reduction techniques applied to other param

eters required for correlation analysis is reserved until those parameters

have been identified.

C. Reduced Data

Operating data are tabulated in Table J. Test values are listed

for the independent variables of heat rate (Qe) ; condenser sink tem

perature (TV} , and sink unit conductance ( c). The operating tempera

ture (To) is also given. Tables 8 and 9 show pressure and typical

temperature data for the water and methanol tests respectively.

7&"

Measured pressures of "both condenser and evaporator transducer

differ from the saturation pressure at the operating temperature (e.g.

Test 9, To = uo.15 °F , P*at. - Z.U>\ s*. Wo)) . A probable cause for

these differences is that each transducer is positioned in a region

where temperature differs from the operating value. The condenser trans

ducer measures a pressure corresponding to a saturation temperature

which is "between the condenser wall and operating temperatures. Simi

larly, the evaporator transducer measures a pressure corresponding to a

saturation temperature "between evaporator wall and operating tempera

tures.

Figure 13 illustrates typical axial temperature distributions

for, water (Test 9) and methanol (Test 20). Near isothermal conditions

are observed along the adiabatic shell wall. Vapor temperatures for

Test 9 coincide with values along the adiabatic wall. The condenser

vapor temperature of Test 20 agrees with the condenser wall value,

suggesting negligible radial heat transfer near the axial position of

thermocouple T3 . High evaporator wall temperatures (Tu*) a r e

observed for both tests. This suggests that wick drying has occurred.

If drying exists, it must be a partial drying, since evaporator vapor

temperatures are nearly equal to the corresponding adiabatic wall

temperatures. For high heat rates (e.g. Tests k and ll) increases in

the evaporator vapor temperature (Tvi) over the adiabatic wall tem

perature suggested a large amount of drying, and testing was terminated

to avoid damage to the heat pipe.

77

Tat>le 7. Reduced Operat ing Data.

F lu id Test Q e T c O c ' To

1

BTO WR °?

STO

Vtf. FT1" ° p . •F

Water 1 155.0 ' '• 69.74 236.46 93.55

2 230.3 70.43 234.89 102.37

3 331.0 71.30 235.25 1 1 1 . 1 1

V " 339.6 72.13 219.67 118.14

5 172.7 70.33 243.23 97.29

6 2^9.9 71.15 232.48 103.82

7 3^9.9 72 .51 220.86 114.56

8 167.5 70.98 198.70 100.57

9 2^7.9 72.90 176.19 110.25

Methanol 10 52.2 67.67 135.61 93.09

1 1 90.8 70.30 133.91 104.26

12 . 39 .0 66.70 229.62 88.54

13 50.8 67.03 183.53 91 .91 i 4 68.2 68.07 156.36 97.87

. 15 - 93 .7 68.77 146.75 103.72

16 52 .7 . 54.66 121.28 85.97

17 71.9 57.6o 116.25 93.10

18 80.8 60.44 l4o .43 100.29

19 24.9 46.95 202.59 73.64

20 6 6 . 1 48 .81 138.85 86.07

2 1 20.6 42.97 186.66 71 .51

Pressure

Table 8. Axial Profile Data Sampling for Water*

Temperature

cond. evap. UOH- n o ^

vapor prohe

T3 Tv.

condenser s h e l l w a l l

TV TT.<O "Tri ~^7A

adia"batic s h e l l v a i l

Tio T» "f«-

• w i c k

*<k " T T H -

evap, s h e l l v a i l

Xvv3

Test : MHg\, irbll/g;

1.21 2.13

J~:?? °? »p op op Op op op op of op *F

1

: MHg\, irbll/g;

1.21 2.13 93-99 9^.08 82.86 80.97 84.06 81.87 94.00 94.00 9^.00 8 o . 4 i 93.55 147.77

2 1.66 2.66 103.12 102.79 88.59 84.07 89.30 86.42 ioi.4o ioi.4o ioi.4o 86.26 102.37 210, 90

3 2.25 3.33 111.74 I I I . 5 6 94.36 88 .97 94.17 90.42 111.39 111.50 111.50 93.28 111.11 308.97

4 2 . 8 i 3-9^ 118.96 126.09 99.28 92.14 97.16 94.52 117.76 117.58 117.99 100.75 118.45 377-10

5 1.32 2.63 97.70 97.28 85.39 82.82 86.05 84.30 97.12 97.36.. 97.36 83.18 97.29 i 7 4 . l l

6 1.68 2.90 10^.51 103.75 88.90 86.14 90 .01 87.80 104.42 io4 .42 104.42 90.12 103.82 247.20

7 2.35 3 .67 114.42 113.47 96.82 90.93 95.6o 94.73 114.73 114.73 114.83 98.26 114.56 3^2.45

8 1.58 2.80 101.12 101.79 88.29 88.07 89.53 88.77 100.78 100.78 IOO.89 86.o4 100.57 169.83-

9 2 .19 3 .58 110.57 110.09 95.24 93.74 95 .71 9^.87 110.54 110.21 108.40 96 .31 110.25 210.50

• Thermocouple numbers refer to locations shown in Figure 11.

•-r

Table 9. Axial Prof i le Data Sampling for Methanol*

Pressure Temperature

vapor probe

condenser shel l v a i l adiabatic shel l v a i l

wick evap. she l l wall

cond. Pxo*

evap. ?xo3 V Tk 1 .5 n.<o , Tin T^.6 Tio ~fw TVx T, T\io TivV

Test in Hg in Hg *F °F «F °F dF "F "F «F °F °F °F aF

10 8.30 9.70 71.07 96.83 73.51 76.35 79.1^ 77.91 93.34 93.24 93.24 79.61 93.09 130.32

11 11.48 13.21 102.26 123.90 84.04 82.51 84.89 82.50 104.56 104.56 104.56 87.69 104.26 199.00

12 7.22 8.57 67.57 90.67 68.43 71.73 74.96 73.86 88.22 88.22 88.22 73.75 88.54 114.70

13 8.05 9.46 69.97 95.44 71.82 73.09 76.78 75.23 91.78 91.78 91.91 78.02 91.91 130.22

14 9.42 11.05 85.64 103.11 79.74 76.08 79.50 77.49 98.06 97.79 97.79 83.18 97.87" "157.95

15 11.11 11.53 100.22 II8.IO 82.53 79.99 82.70 80.37 103.98 103.63 103.63 86.16 103.72 196.79

16 6.70 7.95 59.48 89.51 61.71 67.00 70.30 68.81 85.55 85.57 85.77 68.90 85.97 123.14

17 8.26 9.78 77.11 98.86 72.16 71.39 74.49 72.74 92.56 92.56 92.45 76.80 93.10 154.31

18 10.19 11.94 95.22 118.10 76.81 77.05 80.26 77.36 100.33 100.33 100.33 80.39 100.29 195.25

19 4.61 5.43 47.63 75.59 48.21 49.58 57.31 57-53 73.87 73.87 73.87 54.89 73.64 86.82

20 6.79 8.06 60.14 91.58 62.05 61.95 65.90 63.41 88.00 88.22 88.22 68.10 86.07 114.58

21 4.35 5.08 43.62 73.60 43.29 44.89 52.79 52.90 71.58 71.81 71.91 50.87 71.50 85.25

•^Thermocouple numbers refer to locations shown in Figure 11.

SYMBOLS'.

O WATeR,SrtEU_WAU_") E S T ^

Q WATER. , VAPOR J

I~| METHANOL . SWEU. WALL^ l - J ' S-TeSTZO ^ METHANOL , VAPOR J

lc JL

O -

CONDENSES

1 I 4- -r ^llf Tic* T> ItT '*

t * TTo

ADIAfcATlC

I Tn T i \ r

- < S > - o XT

O •D — O

C] D S

EVAPORATOR

i i Tii- "^\3

SH D

JSL

8 lO 12 (WCHfcS

IH- \4> \8

Figure 13. Typical Axial Temperature Distr ibutions*.

•^Thermocouple numbers refer to locations shown in Figure 12.

8l.

CHAPTER V

CORRELATION THEORY

1. Objective

The purpose of this analysis is to determine the interrelations

between the various heat pipe parameters. This study will l) state the

governing simplified equations, 2) identify the parameters required for

correlation, 3) demonstrate the data reduction techniques for evalua

tion of experimental values of the parameters required for correlation,

h) provide interpretations for distinguishing between internal inde

pendent and dependent parameters, 5) use the correlations to predict

solutions for untested conditions, and 6) compare predictions to experi

mentally tested conditions.

2. Overall System

The structure of the heat pipe consists of a thin walled cir

cular cylinder and a multiple layer of wire mesh screen lining the :'

inside of the cylinder. A working fluid is distributed within the pipe.

The structure is divided into the three axial sections of condenser,

adiabatic, and evaporator. The structure is divided into the two radial

regions of vapor and liquid. An external region surrounds the liquid

region. See Figure Ik.

The overall system is studied by considering a pressure balance

over the entire cycle of evaporation, vapor flow, condensation, and

liquid flow. Using the approach of Kunz, et al, {lj, 18) the pressure

82

v,e

*,«

L e

EVWORM-O*

V/M»OR

\_\QO\p

JL f\oi*e*Tic

«V

*v U

CON06NSER

Figure 1^. Correlation Theory Heat Pipe Schematic,

"balance i s made from Figure 14 "by writ ing the iden t i ty

(fV-Pv,e) 4" OV-Pv^ + (V^c-Vl.Z) V {?A,c-?JLt^) - O . (V.l)

The first term is the pressure drop across the liquid-vapor

interface in the evaporator section. Neglecting inertial forces, see

Appendix L, and characterizing the meniscus "by a single radius gives

(fre-Pv,i) = ~ % y ;(V;2)

where Vw is the meniscus i adius at the evaporator end of the pipe.

The second term is the pressure drop along the path of vapor flow

which is assumed to "be negligible (l7> 18), see Appendix M.

The third term is the pressure drop across the liquid-vapor

interface in the condenser section. This term is assumed to "be. neg

ligible (8, 17> 18) which implies this interface ,to "be a flat interface.

The fourth term is the pressure drop along the,path of liquid

flow. Using Darcy's law for flow in porous media (17/ 18, 27) gives

(?A,I-?X,1) - K 1^%U.^iw«) , (V.3)

where Ki is the wick friction factor, see Appendix F, Aw is the

wick cross sectional area, and i.e{fec-twe is the effective frictional

length of liquid flow. Equation (V.3) is simplified "by assuming mass

flow to "be uniform and radial at the liquid-vapor interfaces of "both

evaporator and condenser (8, 1J, 18). It is further assumed that all

heat goes to phase change (5, 8, 17, 18). These assumptions provide

the simple relations

m

JLffectiie - ^ L e +IOL + tL t j *«<* (VA)

respectively. The inertial term for axial liquid flow is neglected (lT),

see Appendix 0.

Substitution of Equations (V.2), (V.3)> (VA), and (V.5) into

Equation (V.1) and neglecting the second and third terms in Equation

(V.l) yields

-Z, --TT^^V^*^**-^ • (v.6)

Rearranging and using the pore radius tc to characterize the wire mesh

opening size provides the relation

'ftThvA/""^^ , A A ^ ,

r<- *" Kvrcf-tw-vlc-v^Lc) Js. a e. J i — A ^ . (v.T)

The overall internal heat transfer analysis is made "by identify

ing the various thermal resistances. From Figure, 15 the "balance is

(Tew-Jcw) " /„ Q\

<** = " R ^ T T K I T ^ A •*•** * ^ > (V*8)

where Tew and TCVJ are the average temperatures at Tw along the

active lengths of evaporator (L-Q/) and condenser (Lc") respectively.

85

Figure 15. Correlation Theory Thermal Resistances.

86:

Considering the relative magnitudes of the thermal resistances, see

Appendix P, Equation (V.8) reduces to

Q«. - -^TZ. • ( v-9)

Since the effect of thermal convection in the wick is small (8, 17, 18,

21), the wick resistances are described "by the simple conduction model.

Hence

Q e r ^ ' T ^ . ( V . 1 0 )

_ HI— +-X-ttXe f fL« tTTX«Ff tc

where Xetf is the wick's effective thermal conductivity, and Y"ve and

frc are the radii, from heat pipe centerline, of the liquid-vapor

interfaces in the evaporator and condenser sections respectively. In

order to provide interpretations of the parameters of Equation (V.10),

the condenser and evaporator sections are considered individually.

3- Condenser

The fluid flow path in the condenser is modeled as that of vapor

flowing axially into a long cavity with porous walls. The length of the

cavity is divided into two regions, one with vapor flowing, and one

with the vapor static, see Figure l6. For small axial inertia enter

ing the cavity, the flow is considered quenched, i.e. the vapor ceases

to flow axially "before reaching the design length of the condenser. For

8f-

-*» VAPOR

Zk

DYNAWUC STATIC

AD\A6/\TtC

U AXTWE

CONDENSER

Figure l6. Correlation Model of Condenser Active Length.

er

high axial inertia, the flow is considered unquenched, i.e. the entire

design length is filled with flowing vapor. This model suggests a

relation of the form

U

X — {unct'\OY\ (Re) (V. 11)

where (^/Xc) is the ratio of active to design length of the con

denser, and Re is the vapors axial Reynolds numher at the cavity

entrance. Using Equation (V.5) for the mass flow rate of the vapor,

Equation (V.ll) is expressed as

( x ) - f v ( ^ w f c W ) . (V.12)

Previous investigations on vapor flow dynamics have suggested the

possible existence of an active condenser length. Analytical studies

"by Knight and Mclnteer (15) and White (36) on the dynamics of axial flow

with radial suction indicated singularities or no solutions for various

flow conditions. The analytical study "by Bankston and Smith (l) sug- ,

gested flow reversal could occur and stated "Thus, the character of the

condenser flow depends upon the evaporator Reynolds number or velocity

profile at the entrance to the condenser, the [design] length of the

condenser, and the condenser Reynolds number, even for the simplest

situation where the condensation rate is uniform".

Evaluation of the measured active length of condensation (Lc)

is accomplished "by use of a data reduction model "based on l) radial

heat conduction through the wick and pipe shell, 2) measured vapor and

8%

outer shell wall temperatures., and 3) "the measured heat pipe heat

transfer rate. See Appendix Q for reduction details. Consideration

of the experimental data taken in this investigation suggests the

function f1>; see Figure ±7, to "be

(TQ = l-^5(°-3K*^wi-U0), W (V.13)

2- Qe

£& <C ^ u O ^ ^ H-08 aYjL (V.lM

AJU) " L 0 > ^ (V.15)

^ W f ^ A * 4"'°8 • (V.16)

The liquid-vapor interface of the condenser is modeled as a flat

interface., see Figure l8. Hence

rvc ^ rv . (V.IT)

The heat flow path for the=condenser is modeled as that of

simple radial conduction through the active length of the wick followed

"by heat transfer to the heat sink which surrounds the entire design

length of the condenser,, see Figure 19. The wall temperature "TON I S

interpreted as a self adjusting parameter depending on the chosen

external conditions of Tc the sink temperature and 0 C the unit

conductance "between the sink and the wick outer surface. Hence the

SYMBOLS

O WAT&R

Q M6TKAN0L

• • D

D

0^

^ -1.2*5 [0.3 luCRe^ - t . ° ]

Rev «C 4-o8

2 0 fco t o o m-o 180 2 2 0 ZfcO 3 0 0 3M-0

Qe Rev — -ir \\vJlrv Av

3 8 0

Figure IT. Correlation Data Curve for Condenser Active Length Ratio, and Axial Vapor Reynolds Number.

4-20

91.

Yv,c - Yv

Figure 18. Condenser Liquid-Vapor Interface Schematic,

I

9a:

JU

Lc

A-

ACTIVE

t <

rv

fw

V/^POR Tv = ^ o ' I f . r f

WICK < l ^ C J W

' V

lew

SHELL

v

INSUIATION I

Tc

uck

Figure 1$. Condenser Thermal Resistance Schematic.

93

condenser contribution to Equation (V.10) is expressed "by

/ _• \ • r h^L +~-~±—-(T0 Tc) - QejJ^ouTu U^irr*U

(V.18)

where To is the temperature at the liquid-vapor interface along the

active length, equal to the heat pipe operating temperature.

The effective thermal conductivity is modeled as a combination

of parallel and series conduction through the solid -wick-liquid region.

For a thin -wick consisting of multiple layers of wire mesh screen, each

saturated with liquid, and separated by a thin region of liquid, see

Appendix E, the effective thermal conductivity is expressed as

X< - = TTfc\\ KT ^C^-^c-xr^s) -t-

+• , /r*+U*-*)^A\^3E2\ 4 - W : s—/—ISA]

K 6 )\ lnW^-^(\4-%y

where

9h

k. Evaporator

The fluid flow path in the evaporator is modeled as that of vapor

flowing out of a porous wall (wick surface) into a vapor cavity. Make

up liquid is supplied to the porous wick. The length of the wick is

divided into two regions, one with liquid flowing in the wick and one

with the wick filled with static vapor, see Figure 20. Depending on the

particular wick structure and for a given operating meniscus radius, see

Equation (V.7)> "the liquid filled region retreats axially in a manner

to change the active length (L* This model is studied "by considering

a pressure "balance at the end of the active length, see Figure 21. The

pressure "balance around the indicated loop is given "by

(Pv,-PO -V (FW-Pj^ + (PJL,-JW) -v (PV+-VW) -o . (V.21)

Dynamic pressure drops are negligible since these drops are over a

short distance of order rt. The inertial pressure drops across the

liquid-vapor interfaces are also negligible, see Appendix L. Hence the

terms of Equation (V.21) are given by the simple relations

(^-Pw^ ~- O , (V.22)

(Rh.- PJO -- ^ U , (V.23)

(PJ»» ~ &*) ~~ " L - + , **A (V.2k)

(Pv* -P„.) - o . (v.25)

9$

tVAPORKTOR. (\D\N&bT\.<L

Figure 2d. Correlation.Model of Evaporator Active Length.

96

SECTVON SECTION

Figure 21. Pressure Balance Loop at End of Active Evaporator,

9fo

Substituting Equations (V.22) through (V.25) into Equation (V.21) yields

the meniscus radii relation

UU-* - Uz.3 t (V.26)

Since the same meniscus radius must be present at two different sur

faces, the axial position of attachment, characterized by the active

length 0-e) will vary if the pore size in the axial direction is greater

than the pore size in the radial direction^ iy^) •

This model thus suggests that, for a given wick, the function T\_

exists where

{% - W® • CV.27)

The measured active length of evaporation (L«T) is evaluated by use of a

data reduction model based on l) radial heat conduction through the

active length of the wick, 2) radial and axial heat conduction through

the entire length (JU) of the heat pipe shell, 3) measured vapor and

shell wall temperatures, and k-) measured heat pipe heat transfer rate.

See Appendix S for reduction details. Equation (V.7) is used for evalu

ation of (,f»«/fc") . Consideration of these experimental data for a wick

composed of two layers of wire mesh screen separated by a thin layer of

liquid suggests the function f?, see Figure 22, to be

/! v \ O.I 07

m - o.bSt,(Jfe-l) , W (V.28)

MwJu.1 < Vc\.«J • °.wA (V.29)

0.90

0.85

o.8o

0.7S -

LfL JU

0.70

0.(S -

O.&O —

O.S5 —

0 . 5 0

T 1 r 1 1 r T 1 1 1 1 r

-*— =. O.<o%<o Xe &-0

o.lOT

SYMBOLS

«Q WfcTER

• ryteTHM40L-

J L J L 1 I i i J L \.0 |.5 Z-O z.5 3.0 3.5 f.o H-.5 5.0 5.0 6.0 6.5 7.0 7.5 8.0

."Eire

Figure 22. Correlation Data Curve for Evaporator Active Length Ratio and Meniscus Radius Ratio.

so,-co;

U "" *'° > *or (V.30)

Mtt"Ul >' r*L*ial . (V.3l)

The particular coefficients of Equation (V.28) are interpreted as

applying only to the wick tested in this investigation. Although other

investigators may use the same wick configuration (i.e. cylindrical

wrappings of layers of mesh screen) variations exist in wrapping tight

ness and provisions of liquid gaps "between the screen layers.

The existance of curvature on the evaporator liquid-vapor inter

face implies a radial retreat of the interface into the wick. This

radial retreat is characterized "by the effective wick thickness S«;

see Figure 23. The liquid-vapor interface position as required "by

Equation (V.10) is given "by

n,« = r* - *«, _ • (v.32)

where Se depends on the operating meniscus radius, the detailed wick

geometry, and the fluid contact angle. From Figure 23 and Appendix T,

6e is seen to "be described "by

Se r / Tvj. tV>i Y*. YVn A

rw-r„ - x^ TH, re , T^, ~£, *) „ (V.33)

Typical plots of Se. "based on the geometric values used in Experimenta

tion are indicated in Figure 2k.

The heat flow path for the evaporator is modeled as that of

simple radial conduction through the active length of wick and the ;•

id©

PARAMETER VALUES ("THIS l^>VESTl<»^TVO^^,)

Tyw = O . S t f e m.

rv - o. t t 6 «*• r. = O. OOZ15 »«•

rs =• o. oor^s m.

Figure 23. Evaporator Effective Wick Thickness Schematic.

101

i -r> 1 1 1 1 1 - 1 l l "" •"

T O <t>

\<e/

ir

-4>.5

<t> \<e/

ii i

i/U i / i b s w

i

-

<t>

V n V i y 5*

<

i

-

(o.O

<t>

V

* *

n w ?

i

-

-

W w

-5.5 -Tc - O . O O 27 S "

i r ^ ^o .oozzs " -

5.o -

Wi-rv = o .o^s" r¥-o.-2.«?8"

0 1 O 1

** 1 li 1

^•1

o 1 Vn 1

*l ii 1

*G-I

1 " ra

4-.S • H

4-.o -

3.5 -

3 .0 mm

2..S -

2 . 0 - -

1.5

l.o i i i L r - ^ ^ l 1 1 ' O.5o o-55 O.<b0 o.&S o.lo 0.15 o.so o85 o. O o.35 LOO

YW -rv

Figure 2k. Evaporator Effective Wick Thickness Ratio Versus Meniscus Radius Ratio.

102

combination of axial and radial conduction in the heat pipe shell,

see Appendix S. The temperature XjW of Equation (V.IO) is inter

preted as a self adjusting parameter depending on the chosen power

delivered to the evaporator and the condenser environmental parameters.

Hence the evaporator contribution to Equation (V.IO) is not written in

the form of a thermal resistance, as for the case of the condenser (see

Equation (V.l8)), Tsut "by specification of the independ:eht.;parameter of

power

Q e - QsupplieA *° +v+for*Xt>r . ( V - 3 ^ )

5. Systemof Equations

The relations of Chapter V, Sections 1, 2, 3; and k form a

system of solvahle equations.,

A. Dimensional Equations

The governing primary dimensional equations are .

overall pressure "balance

V ** A e- / ^ ^~rr~—o—rrN ) (v.35)

condenser heat transfer

103

condenser active length

Lc - ( ^ v ^ ^ y } (V.37)

evaporator active length

L< = JULU^]-, : (V.38)

evaporator meniscus retreat

&e. • - f3 (*W;, Xij V«i j tc , Tm , <fi) } an<A (V. 39)

wick effective thermal conductivity

*«f{ = f*Cy.S;XjL,ri,rv,Y^,Y-c,n) . {V.ko)

The governing secondary dimensional equations are material property

relations, evaluated at the heat pipe operating temperature. These

equations are

heat pipe fluid parameter

ft<rV\v*. Ax. = f7(To) , ; (V,M)

vapor viscosity

A v * >Uv(To) (V.i*2)

liquid thermal conductivity

XJL- XJLCTO) (V.if3)

101*-

wick-solid thermal conductivity

xs = y*(T} = *6c$ , «a (v. MO

fluid heat of vaporization

ka= WCT^ , (vM)

B. Dimensionless Equations /

Equations (V.35) through (V.^5) are written in dimensionless

form as

H . fa j jgTalW-') •" L ^ &«J jpa1-

o^at^FM7^) (V.46)

a - - 4^W^^I€mi^^^)}; (vM)

[xj = f»([^'VvSfea3) ) (V.kQ)

fee - u t ^ , <v.*9> IS1 - f,(&W*WW) , (V.50)

105

M--^M,®M,w) , (v.51)

E

M - *stf) , (V.52)

•2Qg ~\ ^ "2- O e , . - r r ^ W ^ l " ^ Vv "ffaC^ J CV.53J

fcq-V^ T T ^ I = ^J^

b d - few / ^^)

C. , Dimensionless Groups

Equations (V.35) through (V.^5) contain 2k parameters. With the

four dimensions of length, mass, time, and temperature, Equa'tions

("V.k-6) through (V.55) contain 20 dimensionless groups. These groups

are listed in Table 10. Each group is assigned a numbered "TT symbol

and interpreted as independent or dependent.

Since the dimensional parameters of wick and shell geometry and

environmental coupling (Tc , *- ) Qe) are independent, groups con

taining only these parameters are also independent.

Since the groups on the left hand side of Equations (V.k6)

through ("V.50) are internal self adjusting terms, these groups are

interpreted as dependent. Since the group "X^/X^f of Equation ("V.51)

is determinable in terms of other dimensionless groups, see alsb;~:.' ;_s.-

io6

Equation (V.19)> the group XJL/X«^ is interpreted as dependent.

The groups of Equations (v.52) through (V.55) contain two types

of dimensional parameters either wick and shell geometry and environ

mental coupling (independent) or material property (dependent). If

particular heat pipe materials are specified, i.e. stainless steel and

water, these groups would "be dependentj dependent due to change of

material property with change of operating temperature. If the heat

pipe materials are not specified, the material properties would "be free

to take on any value, making the groups independent. This latter inter

pretation (independent) is taken for Table 10. This choice permits

general.solutions of the equations for any combination of materials.

D. Dimensionless Equations in Terms of IT Symbols

With the interpretation of independent material properties,

Equations (V. -6) through (V.55) from a system of six independent equa

tions with six dependent parameters and 1- independent parameters.

These equations, in terms of the TT symbols are

-rrl5 = f < U ^ ) (V.56)

Tfib.- f x C ^ /• •"• ( V . 5 7 )

TTn = ^-T^^O^-^s) , Q)

-rr£-rc-, (Vrc,bTT-z <-TT, - ^ i ^ s } } \ o ;

TVt6 => f3(TT,7 ;TT5JTr to;TTSj-nq>) ; ( V . 5 9 )

TT,q = ^ ( i T ^ ^ ^ T T ^ - T r ^ T T ^ } a*<* ( V . 6 0 )

Table 10. Correlation Dimensionless Group Listing

107

Dependency Term. Definition

Independent Groups ^ Tc

•n- - ^

^~ Tc.

rw ••TT»' "KT

Remarks

adiabatic length condenser length

evaporator length condenser length

outer wick radius 'Irapor radius

-rr - £-?3gapor:rradius

condenser l eng th

•n- - r * wick pore radius

vapor radius

TTfe Tw* •wick s o l i d r a d i u s

•wick pore r a d i u s

!T7-- K»TcXc -wick f r i c t i o n no.

"Tr6 = n number of screen layers

TTo = <K ~ V o fluid contact angle

TT4< **-X*

wick solid thermal conductivity liquid thermal conductivity

•tr ^ h ) ! * ^ 1 ^ heat pipe number

(Continued)

108

Table 10. (Continued)

Dependency Term Definition Remarks

Independent Groups -rr -S__5S

^^~ TT Vv^vAv/ vapor Reynolds no,

TV,-., = Qe.

»* " -wciv, A cUcT t condenser environment no,

TTm- -UcJU Xs

condenser Biot num"ber.

Dependent Groups TT - -^

JU active condenser length design condenser length

TT- 1=1

"lte" JU

active evaporator length design evaporator length

TT - — - evaporator meniscus radius wick pore radius

TTi6 = Yw«-Yv evaporator wick thickness nominal: -wickt thi^ckhe s s

-rr - A. liquid thermal conductivity

wick effective thermal conductivity

Tr*. To Tc

heat pipe operating temperature condenser sink temperature

109

TT b - ^ + ~ - { \ + TT+'ttio"^"^"^ W W , (V.6l)

where the functions ^ for i = 1, 2, 3; and 4 are known functions.

fi is given lay Equation (V.13). f*. is given lay Equation (V.28).

is the implicit function described in Appendix T. f is given "by

Equation (V.19) with Equation (V.20).

6. Results and Discussion

A. Solutions to 'Correlation Equations

General solutions to Equations (V.56) through (V.6l) are made "by

specifying values of the independent groups and solving algebraically

for the dependent groups, explicitly or implicitly. For description of

the independent groups governing each dependent group, Equations (V.56)

through (V.6l) are written as

TVI5 = TTis(TYvi) , ( V . 6 2 )

TTib - "n , \ t e (Tr i J Tr z ^^ 5 ( | TT^TT 1 , ; Tr ,^ ; ( V . 6 3 )

TTn = l T n ^ / T T ^ T V ^ T T s ^ T V ^ T T ^ (V.6k)

ir.e, = TTl6(-wl|Tri,-n,,-TV5,"afe;TT^^iTllj-Tv,^; ( v . 6 5 )

TT = ^x^i^z^Sj^toj "TCajfT^ , arU (V.66)

ITxo - TTX0(TH;TT*,TVS,TTfa;TT8y-TOO, T ^ T ^ , i r , ^ ) . (V. 6j)

With the large number of independent variables, e.g. eight for "TTia,

there exist many solutions for the dependent variables. A sample of

110

these solutions is represented graphically by plotting the dependent

group vs. the more significant independent group. Consequences of

variation of the less significant groups is indicated by varying these

groups in blocks.?of two.

Figure 25, T c, vs."'. TT,'iIndicates, the variation rof e-ohifen'ser .length

with vapor Reynolds number. This length increases with increasing

Reynolds number. For high Reynolds number, the length ratio approaches

unity, reaches unity, and remains at unity.

Figure^26, TTife vs. TT»; shows the variation of evaporator active

length with heat pipe number. This length is shown to increase with

increased pipe number and to decrease slightly with increased vapor

Reynolds number. Increases in design length and radius groups and the

wick friction number cause a decrease in the active length ratio.

Figure 27, Tt\7 vs. TV\\j indicates variation of evaporator menis

cus radius with heat pipe numbers. The radius is shown to increase with

increased pipe number and to decrease with increased vapor Reynolds

number. Increases in design length and wick radius ratios and the wick

friction number cause a significant decrease in the meniscus radius

ratio.

Figure 28, TTVQ VS. "TTU describes variation of evaporator wick

thickness ratio with heat pipe numbers. This thickness increases

sharply for low pipe numbers and increases slowly for large pipe num

bers. An increase in vapor Reynolds number produces a slight decrease

in thickness ratio. An increase in contact wetting angle produces a

slight decrease in thickness ratio. Increases in length and wick radius

ratios in addition to the wick friction number indicate an increase in

thickness ratio.

TV,.

l.o 1 1 1 1 | I I 1 ^ ^ J ^ .

o.<\ - -

o.a ^ ^ ^ " ^ i r 4 5 ^ TT,5 (TTa")

-

O.l

^ "- IT O.fe #

—

0 . 5

T T • - ^ - <? e

-

o.* =

0.3 -

O.Z / -

o.l —

o„o 1 1 1 I I 1 1 1 ,

^ o GO too m-o 180 ^ o TTii

ifco 3 0 0 3»+o 38o i zo

F igure 25. P red ic t ed Values of Condenser Act ive Length Rat io Versus Axia l Vapor Reynolds Number.

H H H

/ / ' ' '

\\ /

\ / TT,fc - TT[fo (ir, ^ / ^ / ^ ^ T r ^ i T ^ )

/ irib~" 7£ / - ^ _ fa?^ TTT^

. / " M* Qe

( l i t

3

1

-

/ / ' ' '

\\ /

\ / TT,fc - TT[fo (ir, ^ / ^ / ^ ^ T r ^ i T ^ )

/ irib~" 7£ / - ^ _ fa?^ TTT^

. / " M* Qe

( l i t

^ DEF, J z 3 4-

-

/ / ' ' '

\\ /

\ / TT,fc - TT[fo (ir, ^ / ^ / ^ ^ T r ^ i T ^ )

/ irib~" 7£ / - ^ _ fa?^ TTT^

. / " M* Qe

( l i t

TT, JU JU o.5 0.5 3.0 3.o

-

/ / ' ' '

\\ /

\ / TT,fc - TT[fo (ir, ^ / ^ / ^ ^ T r ^ i T ^ )

/ irib~" 7£ / - ^ _ fa?^ TTT^

. / " M* Qe

( l i t

TTa JU JU

o.5 o.5 3.0 3.0

-

/ / ' ' '

\\ /

\ / TT,fc - TT[fo (ir, ^ / ^ / ^ ^ T r ^ i T ^ )

/ irib~" 7£ / - ^ _ fa?^ TTT^

. / " M* Qe

( l i t

•m JGe. fv

\.l »•«. l . o 2 .0 -

/ / ' ' '

\\ /

\ / TT,fc - TT[fo (ir, ^ / ^ / ^ ^ T r ^ i T ^ )

/ irib~" 7£ / - ^ _ fa?^ TTT^

. / " M* Qe

( l i t

TT5 re n, o.ooS o.oo5 o.ovo o.o\o

-

/ / ' ' '

\\ /

\ / TT,fc - TT[fo (ir, ^ / ^ / ^ ^ T r ^ i T ^ )

/ irib~" 7£ / - ^ _ fa?^ TTT^

. / " M* Qe

( l i t

TT7 K,rc Ac Y\<? »-io* Ho 5 l-io*

-

/ / ' ' '

\\ /

\ / TT,fc - TT[fo (ir, ^ / ^ / ^ ^ T r ^ i T ^ )

/ irib~" 7£ / - ^ _ fa?^ TTT^

. / " M* Qe

( l i t

TTi%

i Qe TT y^>U.vhvA

So 4-oo SO H-oo

-

/ / ' ' '

\\ /

\ / TT,fc - TT[fo (ir, ^ / ^ / ^ ^ T r ^ i T ^ )

/ irib~" 7£ / - ^ _ fa?^ TTT^

. / " M* Qe

( l i t 1 1 1

-

SO I O O 150 zoo ^ 5 0 3 0 0 3 5 0 H oo q-50

TT,,

Figure 2.6. Predicted Values of Evaporator Active Length Ratio Versus Heat Pipe Number.

_Yc_

Qe

TTi oer, \ Z 3 4-

"IT, JU JU- 0 .5 O.S 3.o 3.o

TT t JU A*"

O.S o.S 3 .0 3 . o

I T * rw TV l.l u 2 . 0 Z.o

TT5 O.oo5 o.oo5 o.oio O.OIO

TTi V^.rciLc- l io* Wcf HO 5 MO5

""k •Z. Qe SO iK>0 so n-oo

IOO tso q-oo 4-5 o 500

Figure 27. Predicted Values of Evaporator Meniscus Radius Ratio Versus Heat Pipe Number.

1 1 1 1 1 1 1

3 u.

-

I -*"

c TT l6- ^ ( T T i ^ x ^ ^ i r f c ^ ^ T T M ^ ^ TT - - § S -Hire, - - v-

ftffW ^v2~

l i i 1 l

"<V DEF. i 1 3 4-

-

I -*"

c TT l6- ^ ( T T i ^ x ^ ^ i r f c ^ ^ T T M ^ ^ TT - - § S -Hire, - - v-

ftffW ^v2~

l i i 1 l

•ni 1* JU.

o.S o-S 3 . 0 3 .0

-

I -*"

c TT l6- ^ ( T T i ^ x ^ ^ i r f c ^ ^ T T M ^ ^ TT - - § S -Hire, - - v-

ftffW ^v2~

l i i 1 l

TTV 4 e

Ac o.S 0.5 3 . 0 3 . 0

-

I -*"

c TT l6- ^ ( T T i ^ x ^ ^ i r f c ^ ^ T T M ^ ^ TT - - § S -Hire, - - v-

ftffW ^v2~

l i i 1 l

fT3 r*. Y-V

l.l Ul Z..O z.o

-

I -*"

c TT l6- ^ ( T T i ^ x ^ ^ i r f c ^ ^ T T M ^ ^ TT - - § S -Hire, - - v-

ftffW ^v2~

l i i 1 l

TT5 o.oo5 O.ooS o.oio O.OIO

-

I -*"

c TT l6- ^ ( T T i ^ x ^ ^ i r f c ^ ^ T T M ^ ^ TT - - § S -Hire, - - v-

ftffW ^v2~

l i i 1 l

"Hi •fas n 0.5 o.5 \.o \.o -

I -*"

c TT l6- ^ ( T T i ^ x ^ ^ i r f c ^ ^ T T M ^ ^ TT - - § S -Hire, - - v-

ftffW ^v2~

l i i 1 l

"^7 Kvrc JL lio4" l-io* l-io* |.|o*

-

I -*"

c TT l6- ^ ( T T i ^ x ^ ^ i r f c ^ ^ T T M ^ ^ TT - - § S -Hire, - - v-

ftffW ^v2~

l i i 1 l

TT 4 o O TT -2.

TT

-

I -*"

c TT l6- ^ ( T T i ^ x ^ ^ i r f c ^ ^ T T M ^ ^ TT - - § S -Hire, - - v-

ftffW ^v2~

l i i 1 l

TTT^ 3 ^ ^g SO 4-0O 5 0 4-oo

-

I -*"

c TT l6- ^ ( T T i ^ x ^ ^ i r f c ^ ^ T T M ^ ^ TT - - § S -Hire, - - v-

ftffW ^v2~

l i i 1 l

i • • •

-

50 loo 150 ZOO ZSO 300 TTTi

35o «+oo 4-50

Figure 28. Predicted Values of Evaporator Effective Wick Thickness Ratio Versus Heat Pipe Number.

115

Figure 29, TVv<\ vs. "Tio; illustrates the dependency of the

thermal conductivity ratio of liquid/wick effective on the conductivity

ratio of wick solid material/liquid. The liquid/effective ratio

decreases gradually with increased solid/liquid ratio. The liquid-

effective ratio decreases with increases in wick solid/pore radius

ratios. The liquid/effective ratio increases with comhined increase

in wick radius ratio and decrease in wick/pore/vapor space radius ratios.

Figure 30, "fl-io vs. TTi3; shows the variation of heat pipe

operating temperature with condenser environment number. The vapor

temperature ratio increases with increased condenser environment number.

The vapor temperature ratio decreases with increased heat sink number

(TTi+). The vapor temperature ratio decreases with increased wick radius

number.

B. Comparison of Experimental Values to Predicted Values

Comparisons "between predicted and measured parameters are indi

cated in Figures 31> 32, 33> and 3 « Since predictions for conditions

tested in this investigation are made for specified materials, the full

set of Equations (V.k2) through (V.51) are solved. This requires

solving for the operating temperature prior to evaluation of material

properties as expressed in Equations (VA8) through (V.51). For com

parisons to the experimental works of others, material properties are

evaluated at the measured operating temperature. This is necessary

since reported data are incomplete, with insufficient information to

permit solution of the operating temperature. Deviations less than +10

per cent "between measured and predicted values are interpreted as ran

dom deviations. Deviations greater than +10 per cent are interpreted

~y XT

^w - "^u ("H"*,"**, TU>,^e ;TV,0)

IT* - **

ITIo =

X«ff

Xs XA

TTI OE.F. i Z 3 v

TT-3 Yw

i - l- 1.1 Z.O ^.o

ir5 Tv o.olo o.oio o.oo\ o.oo\

TTfe Ywi O.S l . o O.S uo

TT6 v\ Z ^ -2. Z

20 30 iVO TTV,

50 <s>0 IO &O <*0

Figure 29. Predicted Values of Wick Effective Thermal Conductivity Ratio Versus Wick Solid-Liquid Thermal Conductivity Ratio.

— I — I — I I I II11 1 — I — I II 1 II

TT-ic- TTlo (^IT^irs^^lVa -trvo | , TT» ; 1 \ ^

r — i — i i i i 1 1 T 1 — I I I I I I

TT,-

To = Tc

- Qe TT,-* - -nrr^jl, ,U tTc

• ^ OfcF. \ -i 3 4-

~ 3 n* \ . O l l.ol l.io l .\o

TT* ft JU O.Ol o.ot o.io 0.10

TTs Tc

"n" O.OOJ o.oot o.o\ o.o\

TT<i, IVKS

Tc o.S o.5 LO l . O

TT6 n Z z z 2

TTio %4 l o 10 8 0 8 0

TT,X Jk_ Qe ^ o o IVOO 5 0 SO

TT,* UciU

5 S o 5 5 0

J I 1 I » I I I

\o -5

J I 1 » l » 1 » \o~

Figure 30. Predicted Values of Heat Pipe Operating Temperature Ratio Versus Condenser Heat Sink Number.

118

S i

l.c*

C l.°8

D UJ

t> o 01 CX

a.

l.o7

i*F I-

o

ofc

1.05

I.on- -

\.oT>

1 — r —

SYMBOLS

O WATER , TWS \NV£STia»ftT\ON

• WETW*sNOL , TWS ltW6ST\CifcT<ON

T

1.03 \ . 0 f \.os \.Ofe \.07 \ .08 \.Ctf

"^zo 1 T /N\e(\suaeo } T^BLE 7

Figure 31- Comparison of Predicted and Measured Values of Heat Pipe Operating Temperature Ratio.

I.o

vS o.ft -

3 <3

U* O.fo

^ OA

vT

0.2 -

o.q o.o o.z O.H- O.C 0.8

"^"»5 ** Jit, JmtftSOR£D ; /\PP6ND^ Q

119

1

5VWB0LS

i i 1

Q WATER , TWS INME5T\e^T10N

Q M&THAMOL. , TWS IrWESTl&M'lON 0 0 WATER , ScaWPOtTt c*«0 0

d^ " 0 WATER; MILLER <* UOUA (Z42)

D

D B / / o

%P 9

Cte/^

" JS y<&

i 1 1 «

-

\.o

Figure 32. Comparison of Predicted and Measured Values of Condenser Active Length Ratio.

120

0.85

r*n >5 1

Z O

& uu

Oi

a>

b s a D-

J *

o.8o

SYMBOLS

Q WfvTER , TW»S INVESTIGATION

• METHANOL, TWSllMVESTJCiMlON

0.75 -

0.70

O.feS

o.fco

0.5S O.S5 O.fcO

± O.feS

Le

0 .70 0.75

Le \ "Nit • j£€ J MEASURED , kppENoi* 5

o.&o o.8S

Figure 33• Comparison of Predicted and Measured Values of Evaporator Active Length Ratio.

• * -

SYMBOLS

Q WM"ER , TUtS \NVfcSTl6fcTlON

• INAETUP MOL , THIS INVESTlCftTlON

O

uJ a.

>

o.7 o.<?

AkPPENDlX U

t.3

Figure 3 » Comparison of Predicted and Measured Values of Overall Heat Pipe Thermal Resistance.

122

as inherent deviations due to uncertainties of the measured heat pipe

heat transfer rate at low operating power levels, or small residue

deposits in the wick and on the wick surface. These impurity deposits

are suspected to "be the result of deterioration of the enamel insulation

of the wick thermocouples.

Figure 31 indicates agreement between measured and predicted

values of "TCLO; the ratio of heat pipe operating temperature to con

denser sink temperature. All of the predicted values deviate less than

10 per cent from measured values.

Figure 32 shows agreement of TTis, the ratio of condenser active

length to design length. Included are data reduced from the experi

mental measurements of Miller and Holm (2^) and Schwartz (29). Sixty

five per cent of. the predicted values deviate less than 10 per cent

from measured values.

Figure 33 depicts the agreement of "n"ife; the ratio of evaporator

active length to design length. Data of other investigators are not

indicated due to incomplete data reporting and employment of wick wrap

ping geometry different from that tested in this investigation. Suffi

cient data is reported by Fox, et al. (10) to indicate that a variable

evaporator length does exist,, although Fox failed to deduce this con

clusion, see Appendix V. Ninety five per cent of the predicted values

deviate less than 10 per cent from measured values.

Comparison between predicted and measured values of internal

overall thermal resistance is indicated in Figure 3^. This resistance,

see Equation (V.9) is given by

123

(V.68)

Evaluation of the measured value of R is accomplished by la-sing the

measured values of the temperatures in Equation (V.68). See Appendix U

for evaluation details on measured and predicted values of the resist

ance R. Eighty five per cent of the predicted values are within 10 per

cent of the measured values.

R -Tevx ~ Tcvj_

Qe

124

CHAPTER VI

CONCLUSIONS AND RECOMMENDATIONS

Results of this investigation provide a "basis for conclusions

and recommendations in the three areas of General Theory, Experimenta

tion, and Correlation Theory.

1. Conclusions

Based on the analysis .of the General Theory it is concluded that,

in addition to the dimensionless groups described or implied by others,

the ratio of the wick-solid radius to the wick pore radius (.XVws/Yc)

may be a significant group.

From the experimentation the following conclusions are obtained.

1) Stable heat pipe operation is possible even when a fraction

of the evaporator wick is void of liquid.

2) The independent variables, in addition to materials and fixed

geometry, are Tc; the temperature of the condenser heat sink, Uc,

the unit conductance between the condenser sink and the outer surface of

the wick, and Qe, the heat rate supplied to the heat pipe evaporator.

Based on the Correlation Theory for the ranges of variables

studied in this investigation, the significant conclusions are as follows

l) Although the axial pressure drop of the vapor is negligible,

vapor dynamics are important. Variation of the axial vapor Reynolds

number effects the active length.of condensation. Corresponding to

these length changes are variations in the overall thermal resistance of

the heat pipe.

125

2) Variation of the evaporator meniscus for different heat pipe

operating points reflects variation of the active length of evaporation.

This dependency should exist only when nonisotropic wick geometrical

configurations are employed, e.g. when pore radii within the wick are

larger than the pore radii on the wicks surface. Corresponding to

the length change there exists a fraction of the wick which is void of

liquid where axial heat conductions exists in the pipe shell. The

length change results in variation of the overall thermal resistance of

the heat pipe.

3) 'Extensions of the active lengths of condensation or evapora

tion into the adiabatic section as introduced "by consideration of shell

or wick axial heat conduction are negligible.

h) The significance of the wick-solid radius, W<s} is reflected

in variation of the effective thickness of the wick, Se. For different

operating points, employment of different values of ^« would cause

different levels of recess of the liquid-:vapor interface into the wick.

Variation of this thickness results in variation of the overall thermal

resistance of the heat pipe. -

5) It is sufficient to model the evaporator liquid-vapor inter

face as a spherical surface of radius Tm .

6) Darcy's law for liquid flow in porous media is a suitable

model for analysis of liquid flow in heat pipe performance computations.'

7) Beginning with the dimensionless groups obtained in the

General Theory, the groups explicitly insignificant include vapor

Prandtl number, liquid Prandtl number, density ratio of liquid to vapor,

Kutateladze number, liquid pore Reynolds number, and vapor Eckert number.

126

2. Recommendations

Future investigations of heat pipe performance should include

studies on l) the range of validity for describing the evaporator

liquid-vapor interface as "being composed of spherical surfaces and

2) measurements of the retreating liquid-vapor interface and the vari

ation of wick thickness with meniscus geometry.

Further testing of the correlation models and conclusions of

this investigation should include use of wider ranges of the variables.

Such studies would include l) working fluids., for high and low opera

ting temperatures, 2) shell materials, for high and low thermal con

ductivities, 3) shell geometry, for variations of thickness and design

lengths, k) wick geometrical configurations, "both isotropic and non

isotropic, and 5) environmental parameters of T c , Uc.} and Qe .

127

APPENDIX A

GENERAL THEORY EQUATIONS IN DMENSIONLESS FORM

Equations of the general theory are written in dimensionless

form "by first expanding the vector equations into the component equa

tions and then using the reference parameters to ohtain dimensionless

coefficients of the dimensionless equations. The spatial "boundaries,

spatial locations, and constitutive equations are written in dimen

sionless form as "necessary. 'Equations are presented for functions

previously written in words (e.g. radius of curvature Rm = Rw(surface

geometry).

A-l. Vapor Region

A. Spatial Boundaries

The equations are

,o < % < K l + bfc] +1 , (w)

o < r < i + n *} <xx\c (A.2)

o < e £ \ . (A.3)

B. Field Equations

The equations are

equat ion of c o n t i n u i t y

K r / ^ O i a(Pv-0 ^(pvoiv^ = ~ +• T ~ : — - +- 2. _ - ^ r=- o

* r -2.-TT ^ e s> ;

equation of motion in radial direction

LAv ^f- 7JT 2 6 o > r J ' *yf~

[^^U^'^Vr) + -v-

3-^-/r f — ^lli^^^L.^- 7 ^i^A . -V 3 ^r^Xv^f "" r 2.TT as ^ ^ \ j)

".+ ^ T ^ l / ^ U >©• ^ ^ > J ^

r«r -»r ^ / SUA.- J L _ / - ^ S A x.

equation of motion.in angular direction

•*" ^TT 3" f e ^ v \ j ^ ? -Z_TT f .© •*• **- y y -t-

, -2- ( - I x L. L , vA \ 4- ? |^v f >§ 4- -— - —J J Jr

equation of motion in axial direction

- r _ GJV . j-_rAti^L c ^ .. ^ J ^ w , ^ - 1 K\y^ Tf " VFUJT 5f + H" l *^TJ = . r s n ' ^ +

r.- -i v ^ / - VkY f M i - i - f s ^ S A . + LXjT^l f^^J+ l"5"J f ar< ^ j +

+ J^-\J±\ J-2. (r. J _ J S A J v rrvH \ * (**•, «v^\.r-j-f ^ o ^ v ? d S / ^ ^ r V x j T T I ^

(A. 7)

equation of energy

ft Wv7? ^ ^ ^ v ^ ^ L

n A N

- Ufr Tvi vWI^Cf X,

(A.8)

H ( * tf) +^U%%) +1$%(%%\ +

-t-(Av

cp,Tc ^ a? I.-* x is <- \ \

130

A-2. Liquid Region

A. Spatial Boundaries

The equations are

iU o *•* * Cx>lx l + » ; (A.9)

*+M*\ < ? ^ t f i ,anJl (A.10)

O < G < \ . (A.11)

B. Field Equations

The equations are

equation of continuity

J_ *&*& _ J_ V«i ^ f-r^rA ^ = O (A.12)

equation of motion in radial direction

equation of motion In angular direction

- \yx *** •^rMzmh % - -fM+ <*•*>

+ '.KibfeHt-H ,

equation of motion In axial direction

--terttMct^lMM £ £L§. J L UU . *nd £o l ft '

equation of energy

(A. 15)

f- *W a * ^r ^"f Ti-^t^M-1)^^] =

_ rrcir-i^Trj^ir wi, - I M L ft u * r t l LM* c?*i L x* J

±_ £_ / - Y - ^ ^ J L J L / - yrA

(A. 16)

-h

132

+ [ £ M ^ §*(q|B>4l .

C. Constitutive Equations

The necessary equations are

equation of wick friction parameter

K " "R7 = K ( L M , I M , 1>J ; wic* tonfujumtionj (A.1T)

equation of effective wick thermal conductivity

K* = % = * * (fe]x4) xs, \ # , R S I , «) . (A.18) at To /

A-3. Liquid-Vapor Interface

A. Spatial Location

The equations are

* M ± ftVft]^ , (A.19)

r = * + ftll , (A. 20)

o ^ e ^ \ .

B. Boundary Conditions

The equations are

equation of continuity

(A.21)

1

133

( fftu,1 3 - V Vk 135 _ >3i &±) +

+ (P*V«-feJ'* ' A H 5P »f »W

- ( H ^ % Wl - ° •

no slip conditions

(&-*}$ - ( ^ ( M S ^ + ^ " H * - ^ * = ° ' <"-23>

(*. - ftSHfi*) * + (*•- M*) t + li-85)

^wi^'-imm^-0'^

13

c- -mm*) w - (*> - USIBH $ + <*••«

equation of motion in radial direction

Htii-XftSf-^wX^M^lHs^ >''• - (A.27)

-ftfcTfcl?. --l^TftrfeMtl +

*tim$ *w - aaTtai^a «~] + • ($$-$ '#) (?-*- - BSTKIM.*.

+

+ UTVUVJUJ N)„ " | _M L^JLl^fcJ T VeJ +

+ LM^LJL"^]^)^ ~ L"tut"lbrllp*^]^ $ +

^ T l f T b f e . " ! * ' ' ( ^ - ° ,

135

equation of motion in angular direction

04^(^^-t*){^^- l^T[ l lr^ + (A.28)

+ L^^UvlLM ^ V e ""UviiwjL6LlLA.rc.uJ T^ r d T *

+

-IKTffi]?. - 4^MT feM-fc] +

+ U r ^ J U - J ^ ) ^ - Lft'^J LfitJLP rcuJ H)6e\ *

+L"^VA4 LM ^\» - ift^r' CM u ^ i ^Xe \ +

+ l r a til K i J <>> (vs(r) a = O i

equation of motion in ax ia l direct ion

+ "t Uc] I ^ M !>] 1 ^ - li^T] tfcl LMcUfll T0rVj| "*"

+ (t t - t [ -IMH^W^^*

- p. - ftfeTK)* - ^[telSfe]*Sl *

-V- L^TuTf^J LrT] ^ ^ > ~~ L ' P - ^ 1 US'] LWiVcl ^ ^ j *•'

equation of energy, with substitution of equation of continuity

(••KTi.X**- t t ) { ^ ( l ^ . ^ ( — l ^ X t ^ + (A-3o)

4-

r^ia^-Tf^Tx (*M 4. - LfrrviM L^cpvJ L" ^ J ^v V^r5J^_ ( v ^ ^ y "*"

^M(&;m -immi&li UL) ^-^)(sai^ -t tftTbsa*- :

-v-

4-

4- K^ (^([^TO % - mmmb ^

+

+ A ^'rv

* ( f t - f %)WU^*k(a-l&S<>i) *

utf -

+P&1[s!fcl U/cfJ?] L~u£ j L f ^ l l f l l ( \ ^ V x A ^ ^ . . ( v j ^ •v

138

4(av-iiI)/[ftTM:«)l'tl T,)n ~ \ftu7j tS] [S^-lf T*)r<>} T

+(-Ov-^([-^l[^l H* - tf^£ [il LP^^iT t U ) +

+(*.-*.) M l e l X ^ T O ^ - HSfftfl'tsS?] t H^)}+

•K^Btf iff*) -H4f)(^-[gT[|Y^l^) '

139

+ (^--^($m ^, - Ma'&gft \ % ?x l ^ -V-

+ (xr* - ^ ^]\rc\^\s - [j^[f[ [^t^j; V)^j +

H uJv-w^(t\^]](^avrv]V L W L S ^ ^ l PA ^ +

+ a ""itSS tgltefec] ("^-(^ - vi fA' , p » < - o

C. Constitutive Equations

The necessary equations are

rheological equations

*v)rv ~* M v } " ^ ( U l ^ ) 5f< ' * (A-31)

1^0

r _ ^vje© Tv'*6 =

1MI**5 -_ _. *. ( (^ ) t + f f l f c^ r r ) - (A-32)

2 -

- ^ M M%]y i^ a((\*t^i^) -V

3-\Tv

* U*m ^ -v- ^ LxvJ a% V

TO 51

:>!> ^•^v^uiy-sf; - - *v (A.33)

/-J—A ^ H ^ U J j /

H*

% ) re

OTv/

- - M< (-i%i y^» Is, A- (A.3M

M 1 -vtelnA*** l * ^ %

^ - ^ e> -

©*> ^

^Vx, (A.35)

TT = - ^ ' ^ /^vLUX

- - >av< Z risTI 1 ^ [rv] *rs

-V (A.36)

- Ti) r r - a " T*\v - T^uTT "" -'LMi ^ , (A.37)

-* (S) - - 4 f c^)^^(^S , (A-38)

equation of latent heat of vaporization

W = l ^ w - [^ f )w , (A.,3)

unit normal vectors

lA2

n-, - - YVN - r A / (or^wm^-h - H^WIIWA-U)

+ H a ^-v'M(4%T^|%-(^?t]; where

0. -

uni t tangent vectors

.o-'k t ,

^rtd^id] ,^

1 3

two-dimensional surface orthogonal coordinate system parameters

F = FC^.M,-^) , CLY«* (A.48)

1, -" %l^M>-i) , (A. 49)

r ad iu s of cu rva tu re (Rm.)

ffi = T '**.[ t(^vo^(^\(ig}[c*)- u.50)

(V) (sf-^ - n nj + \}*\m*)-sf I H I T "

t i •*-

- M ( t t - * ^ 1 +(c^-

-0«?tWW)]{(-[^^( s i t 5>i a-? >f

Ikh.

Hr

>P 5P

/ rc i -^ ih. i&. -V- ^ 5 ^VL ; «J^»

f 5I»-304 _ ^ 55tT\. , (.. \xj\ = \ ^ i / * b - Sit _

i lii- + H^m^ ^h~- ^Si. ,:

2 ^ V ^ 3* „iii5iiV 1

_J

surface gradient of 0"

Veer Vs^r

a:1 iJL^k >p Sf

SsJftfMj1

<^>v •V (^\%lv)x (A-5i>

WM«]) - p H" -v 04*ftp£$ f ^% +

+ ^ 3 > t \

-v-

1 5

+ a? x ****** + ( \ * \ £ U \ *&*St +

+ ^ 5 3> + Hs^ste*

+ 0« r ^ \ V ^ % ( ^ n; vn HW*

4- t o) + ^ ( ^ %((G8 - (-i^Cf +

-t~

r«

-V 0 * » SVS ^It^i Bli?i +

J

>H ii %{m * ("m(SM§ti) -

- V *P ^ v^i^^j *f n * *f ^ )) -V

"•*?*$ { ( ^ + ( ^ K § ^

+ H^T $$*$%)}! ar\<

146

surface divergence of C~VA

^s'S-V; - ^2lLV' -^x\^ $ * (A.52)

4-(.-WSWKSJK^'fc-WS " • f t +

^ W l W 0 - ' ) + (^VMatfW* +

^ ? B S - ^ ' * ^ vo^>&. v x

»

^ 5 n *•— L O ,

1 + ^iMC^K'-Stf

•S|T*$y)(£^ *('*»<)§*> -

* 4 * ! ^ - % - C»©i V3 = \ " ^ \£j

V P *% -V-

• f c ^ C ^ W ^ •*$%••

A-4. Environmental Boundary Conditions

A. Spatial Location

The necessary equations are

radius of axial sections

r -

radii at ends of axial sections

o < r- 4- m angular coordinate

axial sections

X. ° ' > * B3

r*«" m<% - Sa-K ,..J

&MS^ <aa*ffl -V

B. Boundary Conditions

The equations are

147

[%] t (A.53)

(A.54)

O < Q ± \ • afJ, (A.55)

(A.56)

(A.57)

(A.58)

u^v = ° }

liquid equation of continuity for all axial sections

a * = °; (A. 59)

liquid equation of continuity for ends of axial sections

^ A " 0 ; (A.60)

vapor equation of continuity for ends of axial sections

(A. 61)

liquid no slip conditions.for all axial sections

&x - O 7 <x^ (A.63)

' ^ = O ? (A.64)

vapor no slip conditions for ends of axial sections

(A.65)

^ = ° ) (A. 66)

^\/ ~ O ^YN^l

v/

equations of energy for axial sections with substitution of con

tinuity, no slip, and constitutive equations

evaporator:

f fix! »t A ^ = j_ w r^_i _J 0)o " ST ^ Lr"i b ^ f SUTcJ (A.6T)

adia"batic:

d V-

condenser:

equations of energy for ends of axial sections

vapor:

\

liquid

C. Constitutive Equations

The necessary equation is

Xett

1 9

= o , ^ (A. 68)

feVltl- _ U c r v l (l rg9%;

•*«w7? H<*§ •- n U f f h \ UcCT-OivAs, andi (A.69)

'"-'Ma feVfel

"TJ ~ ° , <™<* (A. 70)

J E LTT - O ' n ' (A. 71)

XG^ " Xeff atTe ' X ^ U ^ l X* j 1*1 , £ 1 , fcl , * ) . (A. 72)

150

APPENDIX B

VACUUM IEAK TESTING

A. ,Purpose

The purpose of the vacuum leak testing is to insure that only

negligible amounts of room air leak into the heat pipe during operation.

B. Procedure

The procedure consists of evacuating the heat.pipe over a twenty-

four hour period, closing the valve to the pump, and recording the

Alphatron pressure over a thirty minute period.

C„ Typical Results

Typical results are

date: '7/31/71

room conditions:

pressure response: Time ^Pressure (minutes) (microns Hg)

0 9.0 1 9.2 2 9.6 3 9.8 h 9.8 5 10.0 6 10.0 7 10.1 8 10.2 9 10.^ 10 10.6 15 11.2 20 12.0 25 12.6 30 13.^ 35 14.0

151

initial leak rate:

from Figure B-l

initial leak rate = 0.1^ microns/minute

projected accumulated air:

for 8 hours of heat pipe testing

final pressure = 67.29 microns Hg

-3 = 1.3 X 10 psia at 8 hours

From Reference (6), this is a negligible amount.

U IC1 Z3

TiN\fc ~- WMNores

Figure B-1. Vacuum Leak Test,

H vn ro

153

APPENDIX C

EVALUATION .OF FLUID INJECTION VOLUME

A. Purpose

The purpose is to estimate the volume of liquid necessary to

saturate the heat pipe wick. This amount is to he injected into the

heat pipe during loading of the working fluid.

B. Procedure

The procedure consists of estimating the wick porosity, evalu

ating the liquid volume; required to fill the wick with liquid, evalu

ating the liquid volume required to fill the vapor space with vapor,

estimating the liquid volume required to fill the pipe attachments with

liquid, estimating the volume of wick thermocouple wires, and using

these estimates to evaluate the injection volume.

C. Results

The results are

wick porosity (see Figure C.l):

V S QUO

e - \ - -w V TOTAL

, ZT (sffi (A. *y*+£) VsoUD ~ ^ kw<S X \ N S \ ^.JrX^

VSOMO - ( 4 - N ) ( T T S ) ( . O O T . ^ 5 N ) ( . ou«+ \3 j ( 3 5 . 1 0 ^ IO1" J V?.

CROSS SECT\ON

p ^ V * ^

uNtr -—

[

}

PLAN U^/S*,»VJ

FOR & UlvHT :

V S O U O - 4" j^vNS ^>NS

iws - (zCrc^r^T" + ( Z T c T

15^

Figure C-1. Schematic Diagram for Estimation of Wick Porosity.

VTOTAU = TC (JLc + J L a - v . ^ ( r * - t f )

e - i• - 0 . 2 . 5 ^ - o . i *

wick l i q u i d : ^ A " £- ^TOTAU

V j L = \.1*1)(^\ <*.[<() CC.

V 0 ~ l l - O O CC.

l iquid for vapor space vapor:

v - 2L V

V v = IT (JU+-Jl"* + Jlc>)

evaluating at 120 F for vater,

v.-(^x)W(.-4(^M V.* = O.OOGSk cc

liquid for filling attachments :

condenser transducer 1.6 cc evaporator transducer 0.7 injection valve 0.5 pipe valve 0. 5

Y* = 3.3 cc

thermocouple volume:

c

v<o«o " - W ( T ) ( U )

where

n is the number of wires,

D is wire diameter, and

X/we, is ^ e average length of a wire

V^*M(?^)0t)(l-**1 cc.

Vsouo -: O.0(o cc.

injection volume:

VJL - sum of liquid volumes - thermocouple solid volume

ViL - 12.00 + .OO656 + 3.3 - O.06 cc

VJL = 15.2 cc

This volume, 15.2 cc, is the nominal amount injected in all tests.

157

APPENDIX D

OPERATING DATA

The operating data consists of Qg,the power level delivered to

the evaporator heating costs, V ; the volume of cooling water collected,

and &t, the time lapse during cooling water collection. For the data

indicated in Chapter IV, Ta"ble J, the operating data are

Fluid Test Qt V at

watts liters .. minutes

water 1 48.3 O.858 2.00 2 7 .0 0.816 2.00 3 108.0 0.822 2.00 4 133.0 0.839 2.00 5 54.2 0.800 2.00 6 79.0 0.798 2.00 7 112.0 0.800 2.00 8 52.5 0.597 3.00 9 79.0, 0.590 3.00

methanol . io 19.3 0.850 8.50

n 4o.5 0.647 6.50 12 13.3 x0.796 4.00 13 19.6 0.792 4.00 14 29.0 0.782 4.oo 15 4l.2 O.786 4.oo 16 19.6 0.396 8.00 17 29.9 0.392 8.00 18 4i.8 0.390 8.00 19 5.3 O.632 7.00 20 28.1 0.8o4 8.00 21 5.4 0.810 4.oo

158

APPENDIX E

PRESSURE] AND TEMPERATURE EMF DATA

The emf data consists of thermocouple and transducer signals

recorded during heat pipe operation. Recorder data (2- -6) are given in

units of fractions of 5»000 mv as registered on recorder chart paper.

Potentiometer data (102-115) are given in units of millivolts as

measured. Tests omitted from the data listing are not indicated due to

l) incomplete data recorded during test, 2) operation at too low a power

level for reasonable accurate estimation of Q .- (see Appendix I), and

3) operation at too high a cooling water flow rate for reasonable

accurate estimation of cooling (see Appendix G). Thermocouples

omitted from the data listing are not" indicated due to l) suspected

thermocouple failure (e.g. insulation deterioration and weld detachment),

and 2) considered extraneous (e.g. relay skip-thermocouples and pre-heat

temperature of injection fluid).

Thermocouple numbers refer to the locations indicated in Figure

11, Chapter IV. The pressure and temperature emf data are tabulated in

the following list?;

159

ON

0 0

t -

U VO

<D H

I O LfN O (D

EH

-d"

0 0

LfN

OJ

rH - d "

; 0 J

- d "

OJ

fr-o cn

vo OJ OJ

LfN G \ \s\ OJ

b -ON OJ

o - d -OJ

o 00 OJ

ON O OJ

CO - d -OJ

on CO H

OJ

o OJ

vo OJ OJ

LfN

o - d -OJ

LfN H vo r-H

LfN vo ON rH

LfN

OJ

o

9 CO LfN rH

LfN OJ CO

O i o : o o p O O o o O o O O O o o O o o o O

l > -O OJ

ON cn OJ

00 vo OJ

rH o cn

ON H 0J

0 0 - d -OJ

r-(?, OJ

o cn OJ

LfN t~-OJ

ON ON rH

LfN OJ

- d -OJ

LTN O

VD .H

UN ON

<s O OJ OJ

-d-cn OJ

LfN ON

- d -rH

LfN

-3 rH

•3

LfN ON vo O

LfN OJ cn rH

o LfN O

O o o o o O o o o O o O o O o o O O O o O

LfN O OJ

o cn OJ

o LfN OJ

UN

o CO OJ

LfN CVI 0J eg

LfN ON

-3 -OJ

OJ-00 OJ

-d-cn OJ

H l> -OJ

ON VO rH

CO H OJ

vo LTN H

-d-vo H

l > -l> -rH

LfN ON rH OJ

LfN H rH

CO cn rH

LfN ON vo rH

LfN ON vo O

- d -ON O

o LfN O

O o o o o o o o O O O O O o o o o O O O o

ON H

rH rH OJ

rH cn OJ

OJ LfN OJ

LT\ LT\

o OJ

ON •CVI CVI

LfN LfN OJ

VD H OJ"

vo LfN OJ

o\ H

OJ OJ OJ

LTN

v~-iH

00 H

O

o OJ

ON

o OJ

c — -3-H

CO vo H

o CO rH

c — ON O

vo OJ H

vo l > -

o O o o o o o o o O O o O O o o o O o O O o

LfN O OJ

H cn OJ

H LfN OJ

LfN 0 0 OJ

o rH OJ

i r \ H CO CVI

H l> -OJ

o 0J OJ

o vo OJ

- d -ON rH

LfN l> -rH OJ

O VO H

LfN H CO H

cn o OJ

LfN

OJ

H - d -rH

LfN OJ l> -H

H l > -rH

ON vo O

LfN

OJ H

o LfN

o O o o o o o O o o O O O O o o O O o O O o

LfN - d " OJ OJ

UN rH LfN OJ

UN

OJ

H o on

vo cn OJ

CO LT\ OJ

O ON OJ

o LfN OJ

LfN OJ oo OJ

- d -

o 0J

LfN OJ cn OJ

o ON H

LfN l> -ON H

LfN

o H OJ

LfN OJ OJ OJ

LfN O vo H

LfN rH

-3 o O OJ

OJ H H

LfN O cn H

l > -ON O

O O o o o o O o o o o o O o o O O O O o O

LfN

OJ

on H on

LfN cn LfN cn

LfN t~-

0 0 on

o\ 00 OJ

LfN ON H on

vo vo cn

-d-o cn

00 -d-cn

LfN rH l> -rH

ON o cn

UN LfN LfN H

l> -vo rH

LfN vo cn OJ

o o cn

LfN ON rH rH

ON ON H

CO l> -OJ

LfN l> -vo O

LfN OJ OJ H

O LfN

o o o o o o O o o o o o o O o o O O O O O o

5 UN LfN LfN LfN LfN LfN LfN LfN

i6o

MD OJ

LTN OJ

. VO U H a;

r ^

a; H

o-d-O H O

a; E-i-

a

LTN rH OJ

o r o OJ

LTN rH LTN OJ

MD OJ

OJ OJ

O -3-OJ

OJ MD OJ

LTN

-3-OJ

-3-

OJ

J -ON rH

LT\ OJ OJ OJ

-4-t -H

©• CO. H

CO ON H 21

05

CO LTN

H

LTN OJ

H

LTN

ON H

MD

o

LTN O CO

H

LTN LTN O

O j o O O o o O o o O O O O O 6 o O O o o O

I , , 1

, ,. , -3-LTN OJ

LT\ CO OJ

LTN rR- O

CO OJ

ON LTN

H

^t-C^ ON

O OJ

LTN OJ OJ OJ

LTN ON OJ H

LTN LTN

't— H

MD ON H

o o

rH CO H

LTN

CO -3"

o o o ,o P . O O O o O O O o O o

r o

OJ

LTN H H r o

LTN OJ LTN CO

CO OJ CO

OJ

MD H CO

OJ MD CO

o CO

-4-co

LTN CO OJ

CO H

-=h

CO LTN

OJ

ON

OJ

CO H CO

-3-00 CO

CO LTN OJ

-=h ON OJ

-=fr 0 0 CO

OJ ON H

LTN OJ MD OJ

CO CO H

o O O o o O O o o O O o o O o O O o o O O

1 . , • 1 1

, , 1 , 1

1 :o Oil

LOi OJ OJ > CO

ON -d-OJ

LTV -4 i

vo OJ

ir\ OJ ON OJ

o OJ CO

ir\ t -co OJ

O

OJ

_3-;e CO

ir\ OJ CO H

CO CO OJ

CO

O o o o O o o o o O O o

OJ

LTN

H r o

-=h LT\ CO

LTN C O 0 0 CO

LTN ON CO OJ

o OJ CO

O

CO

LTN O CO

o LTN CO 1

, 1 1 1

, , , , , , , ,

o O O O O o o O O w

OJ

OJ

O H r o

LTN O N

- 3 -CO

LTN O

C O CO

CO OJ

CO H CO

LTN LTN

MD CO

LTN OJ O CO

O -3-co

ON VO OJ

ON H CO

MD -d-OJ

LTN OJ VO OJ

ON CO OJ

LTN

-3-

cn

LTN CO OJ

LTN VO OJ

o o CO

OJ CO H

MD -4-OJ

LTN CO

H

o O o O o O O O o , o O O o O o O o o O O O

OJ

OJ

O H C O

LTN ON

-=h co

O CO CO

CO OJ

CO H CO

LTN V D C O

OJ O CO

LTN 3* CO

ON MD OJ

ON H CO

MD -4-OJ

OJ MD OJ

ON CO OJ

LTN -=h H C O

LTN -=h co OJ

LTN LTN

\D OJ

o o CO

OJ

9 vo -4-OJ

CO

H

o O o O o O O O o O O o O O o o O o o o O

LTN LTN

l 6 l

MD C O

LTN CO

OJ CO

i n CO CD rO

0 H

o o O CO

o

CD ,3 EH

O N OJ

0 0 OJ

0 0 LT\ H 15

75

LTN 00 -LTN H

o MD H

CO

t -H

CO

H

SO

H

so CO H

0 0 CO H 1 l

1 1 1

1 I 1

LTN 0 0

51 OO OJ H

LTN CO H

OJ H

OJ H

LTN OO OJ H

o :;o O O O O O O O O O O o O o

H

OJ

O N

o CO

fe^ -4: co

O 0 0 CO l

i r l 1

ON so OJ

o OJ CO

MD -d" 0J

CO

so OJ

LTN t -

OO OJ

LTN H CO

LTN LTN CO OJ

LTN MD OJ

LTN H O CO

LTN OJ OO H

MD CO OJ

t-

O o o O o o o O O o O O O o O O

LT\ [ > -

MD OJ

OJ

o CO

O -d" .co

b [ > -CO

I l . i

; , 1 I

1 1

LTN OJ so OJ

H H CO

H -d" OJ

[ > -LTN OJ

O OO OJ

o ON O CO

O CO OJ

ON LTN OJ

-d-ON 0J

ON

H

O CO OJ

b -MD rH

O o o o o O o O o O O O O O O O

U A OJ LTN OJ

MD CO OJ

9 CO

LTN

-d-CO

0 0 so 0J

U A CO ON OJ

LT\ 0J CO CO

MD CO 0J

0J OJ CO

-d--d-OJ

LTN CO 0J

-d-OJ OJ

SO CO 0J

LTN LTN OJ

O OO 0J

OO O 0J

LO, 0J CO 0J

CO MD OJ

LTN -d-H

LO, ON ON H

O CO rH

O O o o o O O O o o O o O O O O o O O O O

-d" OJ

LTN

£L OJ

[ > -O CO

o CO CO

o LTN OJ

LTN O N

so OJ

SO O CO

LTN ON

MD OJ

so o CO

CO

o OJ

ON CO OJ

-d-MD H

ON 00 H

O H OJ

LTN ON CO OJ

MD -d" H

LTN ON

H

CO OJ OJ

LTN H [ > -O

OO CO H

OJ LTN O

o o o o o o o O o o • o O O o O O O O O O O

-d-MD OJ

-d-o CO

MD OJ CO

LTN ON LTN CO

CO OO 0J

o H CO

O CO CO

SO ON OJ

o OJ CO

-d-MD H

LTN LTN H 0J

ON -d" H

SO LTN H

o H

H H OJ

-d" O H

H -d" H

O 9

-d-MD O

LTN OJ ON O

MD -d-O

O o o O o O o O o O O O O o O O O o O O o

O OJ OJ

LTN H

-d-OJ

o MD OJ

LTN C O C - -O J

LTN H CO OJ

OO -d" OJ

LTN ON t -OJ

LTN 0J LTN OJ

o 00 0J

0J

o 0J

O C O OJ

-d" OO H

O ON H

O O 0J

CO H OJ

LTN H

MD H

ON t>-H

LTN ON ON H

H H H

LTN

H -d-H

LTN

o ON

o O O o o o o o O o o O O O O O O O O O O o

LTN LTN LTN LTN LTN LTN LTN LTN LTN LTN

Emf Data

Tes t 37 38

1 O.167 0.169

2 0 ? l 6 7 0.1675

3 0.1675 0.1675

4 O.169 0.167

5 O.183 0.179

6 0.186 0.180

7 0.190 0.182

8 0.150 0.1515

9 0.154 0.154

10 0.1485 0.1495

n 0.151 0 .151

12 0.144 0.1475

13 0.144 0.148

l 4 0.1455 0.1475

15 0.150 0.150

16 0.139 0.1395

17 o . i 4o 0 . l 4o

18 0.146 0.143

19 0.135 0.136

20 0,136 O.1365

21 0.1325 0.1325

Thermocouple Number 39 4o 4 i

0.175 0.158 0.2285

O.183 O.1635 0.281

O.1875 0.1675 0.346

0.195 0.173 0.413

0.2095 0.170 0.314

0.216 0.1735 O.387

0.231 0.182 0.515

O.1645 0.126 0.243

0.1725 0.133 O.3085

0.153 — 0.180

O.166 — O.237

0.149 - - O.1655

0.1495 — 0.1795

0.1525 - - O.2025

0.159 - - O.235

0.146 0.127 O.I78

0.150 0.130 0.195

0.1595 0.137 O.233

0.143 0 .121 O.1565

0.1495 0.127 0.194

0.142 0.1245 O.1585

42 ' ^3 44 O.1985 0.170 0.170

O.235 0.171 0 .171

a. 287 0.174 O.1675

0.328 0.190 O.1835

0.215 ; 0.1945 • • _ _ .

0.245 ' 0.1995 - -

0.-303 O.2125 —-

o . i 8 4 0.1505 •- --

0.2225 0.157 —

- - ' 0 .146 0 .151

- - 0.151 0.1575 __ 0.144 0.147

- - 0.148 0.149

- - 0.149 0.150

- - 0.149^ O.1525

0.148 O.1455 O.1305

0.158 0.148 O.1365

0.179 O.1505 •0 .143

0.1355 0.143 0.1135

0.156 0.148 0.1175

o . i 4 o 0 . l 4 i 0.103

Emf Data

Test 45 46

1 0.1815 0.2175

2 0.1865 0.240

3 0.188 0.263

4 0.191 0.2835

5 0.203 0.265

6 O.POQ — * — —

0.310

7 0.219 0.3635

8 0.168 0.222

9 0.174 0.251

10 0.162 0.210

l i 0.169 0.239

12 0.1475 0.1985

13 O.1605 0.2075

l 4 O.163 0.2225

15 O.1685 0.2385

16 0.153 0.1925

17 O.1585 0.210

18 O.1625 0.2285

19 0.1485 0.166

20 0.153 0.196

2 1 0.155 0.206

Thermocouple Number 47 102 103

0.1745 o.o64 0.470

0.1725 0.094 0.521

0.1715 0 .131 0.586

0.170 0.157 0.645

0.1825 0,076 0.496

0.1845 0.102 0.546

0.189 0.1455 0 .621

0.150 0.142 0.525

0.150 0.206 o.6oo 0.1515 0.080 1.195

0 .151 0.170 1.532

0.150 0.025 1.086

0.150 0.039 1.171

0.1485 0.065 1.326

0.1495 0.090 1.505

0.149 0.178 1.029

0.149 0.282 1.205

0.1495 0.405 0 . 4 i 4

0.1495 0.039 O.786

0.148 0.154 1,039

0.149 0.020 0 .751

i o 4 106 107

o . 4 i o 1.695 2.o46

0.584 2.073 " £ 7 5 8

0.812 2.46o 3.483

1.034 2.745 3.986

0,445 - - — - = • - - -

0.595 - - - -

0.866 __

0.556 - - __

0-791, - - - -

3.177 - - - -

4.407 - - - -

2.755 - - - - •

3.080 - - - -

3.617 - - __

4.267 - - - -

2.555 - -

3.159 - - - -

3.909 - - —

1.741 — • —

2.589 - - . . . . __

i . 64o

l6k

l/N

_d-

0 0

0) H

O rH O rH O H

0 rC EH

O

ON O

o _d-ON

ON OJ

- d -OJ oo

MD OO ON

J 1 l

l J J 1 1 ; ! ! 1 1 J i 1 J ; |

H ?0J: oo OO

O l/N

-d" 9 .

H t -H

l/N _d-O

, , I

, ., , , 1 i I

, , , , , I ,

CU CO MD co"

o ON l/N

co OJ

o

ON -.-J-OJ

CO l/N • t -

ON CO H

_=h VD CO

ON CO ON

OJ ON O

o OJ

o

co co H

VD _d-t -

ON OJ CO

VD CO H

O OJ co

ON ON MD

l/N OJ

o

o _d-t -

o t -

MD

o H .CVI

H ON t -H

OJ _d- MD t - - oo _d- MD OO _d- OJ oo H OJ OJ OO QJ OJ OO H OJ H

ON O

- d -

t -VD U"N

O ro ON

MD VD ON

I i I I 1 . 1 1 1

I l 1 I 1 1 1 I 1 i

* • • < • • OJ CO _d- l/N

o oO co

ON CO CO

H ON ON

O O

_d- I I I I 1 1 1 I I 1 1 I

, , , , , ,

H OJ* OJ oo

ON CO MD

l/N ON l/N

CO O l/N

o MD MD

, l I 1 1

, .1 1 , , , , 1

, 1

I 1 i

OJ* CO _d-" L/N

l/N _d-

O LT\

MD

co ON OO

H _d-OJ

, I , 1

, 1

, , , , , 1

1 1

, , , ,

• • • • OJ CO LT\ t -

165

APPEM)IX F

PRESSURE TRANSDUCER DATA REDUCTION

A sample calculation is provided for transducer data reduction

for Test 2 (Chapter IV, Table 7).

A. Absolute Pressure Transducer

emf: 0.584 mv

pressure, from calibration: P = 1.66' in Hg absolute

B. Differential Pressure Transducer

transducer emf = 0.521 mv.

room temperature (TC-Vf) = 71.69 F (from method of Appendix H)

barometric pressure, indicated =29-75 in Hg

barometric pressure, corrected = 29.25 + 0.1l4 (from correction curve)

in Hg

= 29.36 in Hg; abs; '.

transducer pressure differential = 26.39 in Hg (from calibration curve)

transducer absolute pressure = 29.25 - 26.39 = 2.86 in Hg absolute

166

APPENDIX G

POTENTIOMETER DATA REDUCTION

Sample calculations are provided for emf data measured with the

potentiometer for Test 2 (Chapter IV, Table 7).

A. Pressure Transducer EMF

See Appendix F.

B. Cooling Water ;Tempjsrature Differential i .•• , f ' • " "' :

emf = 0.09^ mv. (frem Appendix E)

&T = 4.37 F° (from Figure G-l)

Figure G-l is "based on NBS Circular S6l cali"brations for C^~Cv\}

where an emf of 0.215 mb (k-2 F) corresponds to a temperature differential

of 10°F.

C. Typical Thermocouple

TC-113: emf = 4.028 mv.,

T. = 208.90°F (fronTEBS Cir. 561 tables)

correction factor = 2.00 F (from calibration curve)

T . _ = 208.90 + 2.00 = 210.9Q°F actual

OO O.OCi OA1 O.lft 0.2.*t- O.-iO

EMF ~~ MILLIVOLTS

0 - 3 <o o.*¥L 0 .H6

Figure G-1. Cooling Water Temperature Difference Versus EMF.

CA

APPENDIX H

RECORDER DATA REDUCTION

A sample calculation is- made for emf data measured vith the

recorder for Test 2 (Chapter TV, Table 7)•

Typical Thermocouple:

TC-Vf: emf fraction = 0.1725

T. ,. . , = 71.35°F (from Figure H-l) indicated ^ . & '

correction factor = + 0.3 - F (from calibration curve)

^ a e t u a i * 7 1 - 3 5 * 0 - 3 ^ l i ^

Figure H-l is"based on the EBS Circular S6l calibration tables

for CtA-Ctx with an emf fraction of 1.000 equal to 5.000 mv.

lfaO

150

1 1

IH-o -

130 -

\tO

\ i o

loo -

qo 0-0 I 8o

h —

7 0

(oO y '

5 0 yS

i fo o . 0 0 O.Ofc 0.12 0 . I8

1 J. O.ZM- < x * 0 0."5G O.H-Z

EtAF FRACTION OP S .ooo VWLUVMOUTS

O.H-8 0.54- O.fcO

Figure H-1. Recorder EMF Fraction Versus Temperature.

ON

vo

170

APPENDIX I

HEAT TRANSFER RATE DATA REDUCTION

The heat pipe heat transfer rate data reduction consists of four

parts l) measurement of evaporator coil power/ 2) evaluation of cooling

"water heat transfer rate, 3). estimation of heat rate to the surround

ings, and k) calculation of heat pipe heat transfer rate. A sample

reduction is made for Test 2 (Chapter IV, Table 7). •-..

1. Evaporator Coil Power (Op)

The coil power is that value recorded in Operating Data

(Appendix D):

Q E =V IM-.O WATTS = 152.SO BTu/ttfc .

2. Cooling Water Heat Transfer Rate (Q )

The rate of heat transfer to the cooling water is calculated "by

the expression

Qc = (f^cr^(v)(1^(x) , : (i-D

where

V is the volume of cooling water collected,

bTc is the measured temperature difference "between inlet and out,

at. is the time lapse for water collection, and

ftC.04 is the volume heat capacity evaluated at Tc .

171

For Test 2 from Appendix D,

V - 0.8l6 liters, and

at =.2.00 minutes.

From Appendix 1, the average cooling water temperature is

Tc = 10M0?.

Evaluating material properties ( 0, kk) and using Figure 1-1,

gives Q T O jmv j.

C% Cp^ = \3\ . 8 so uR°R ^T- .

From Appendix E, with the method of Appendix.G,

A T C = *.Z1 °F

Substituting these values into Equation (I.l) gives

G?c - (\*\.**^(O.81^(**T)(JCO). **&. > ^

C?c - ^35.55 V

3. Heat Transfer, to the Surroundings^

Circulation of ventilation air in the room surrounding the test

equipment causes the convective film coefficient (VIJL) to differ from

values calculated from free convection models. This coefficient Vu is

taken to "be expressed "by the relation

Qs = YuAs iff)5 (1.2)

rj2. a

132. fc

IVZ..4- -

»l re: IVZ.7. H U) D V 2 £ 11.2.0

X

3 \ * fr. 1* CD1* !3'..S

i Q_ PM.fe

U c^

13\.M- -

131.2. -

13! .O -

\30.8 loo

T ~ « F

Figure Tr,l. Cooling Water Heat Capacity Versus Temperature.

H -<] ro

173

•where

Q s is the heat transfer rate to the surroundings,

As is the area exposed to the surroundings, and

fiTs is the temperature difference "between the surface in question and the surroundings.

The coefficient (h), used for final data reduction, is taken as the

average value of h. for a series of n tests, where & 1 . ^

W-- i i Yv A=l (1.3)

Considering an overal l heat "balance for a single t e s t , gives

YW

where m is the number of surfaces considered. Solving Equation (i. -)

for \\X gives

< ? * • - Q c

^ "- T * rr<T ( I-5)

3~"4

The respective areas are

Ap _ = condenser insulation area on the outside of the 3 aluminum, foil cover,

A. T = adia"batic insulation area on the outside of the 3 foil cover,

A. p_ = adiabatic area contribution from the condenser 3 transducer,

17^

Ap = evaporator insulation area on the outside of the ' foil cover, and

A„ ,-,„, = evaporator area contribution from the evaporator 11,ET , , transducer.

The respective temperature differences are taken as the difference

between the surface temperature (averaged over the respective area) and

room temperature. In terms of thermocouple numbers, see Figure 11,

these differences are

f o r AC,I' * T s & | ( T C ^ V T C S 1 + T C 3 8 ) - TCH-7 , (1.6)

f o r AA,i> ^T<; = T (TC 3^ V TC H-o - TCH-7 } (1.7)

f o r AA,CT' A T ; S C TCH-5 - TC 4-7 } (1.8)

" f o r A E, - r *T«> = 3*(TC4-1 +TCH-Z+TcH-3,)-TClK7 }«** (1.9)

f o r \ET' * T * Z T C ^ - TC 4-7 . (1.10)

From surface geometry,, the values of the respective areas are:

A = 267.8 in.'sq.

A A = 138.2 in. sq.

AA,CT = 19*2 in* Sq*

A^ =285.0 in. sq.

VET = 5*3 ln* sq*

Temperatures are evaluated from Appendix E, with the method of

Appendix H.

Sample calculation is made for

and k (Chapter TV, Table 7)* where lv

and Equation (1,6) through (l.lO) are

The calculation table is as follows.

Test 2 of the series Tests 1, 2,

is evaluated from Equation (l.5)>

used for temperature differences.

h. calculat ion table 1

Test No

BTU/hr

Q

BTU/hr

ATs)c,I

in2 R°

A Ts) A,I

in2 R°

AT") s;A,CT

• 2 _o" in R

A Ts)E/I

in2 R°

A T ^ s;E,ET

.2 _o in R

A.( T ).

. ft2R°

h. 1

Q-r-i 1 u 1 \s

BTU/hr

Q

BTU/hr

ATs)c,I

in2 R°

A Ts) A,I

in2 R°

AT") s;A,CT

• 2 _o" in R

A Ts)E/I

in2 R°

A T ^ s;E,ET

.2 _o in R

A.( T ).

. ft2R° hr ft2R°

1 164.80 169.66 - 600.76 - 252.22 30.53 1578.90 - 1.27 5.244 - 0,9591

2 252.50 235.55 - 516.85 17.28 61.06 3612.85 80.61 22.6o4 0.8132

3 368.51 329.96 - 424.91 190.72 72.19 6249.10 108.92 ,.43.028 O.9161

4 ^53.81 404.95 - 284.76 440.17 91.97 8897.70 134.73 64.443 • 0.7703

H - 3 O^

177

Using these values of h . , Equation ( l . 5 ) indicates

V - "if ( o . ^ 5 ^ l - v o . e i ^ l + o . ^ i t u o . n o ^ i j or

1 fctv V\ "= O • 8 (a 5 Vtft~fT ~?°

The r e s p e c t i v e hea t t r a n s f e r terms a r e c a l c u l a t e d from Equat ion ( 1 . 2 )

for Test 2

01^-- - 3.10 " " v ^ ,

j T ^

<?*) CoN,D«*W = °" ©<>? ( - 5V<b .8%) (l4-H- )

Q s ^ o i w n c = ; . P . B f c S ( n . l 8 + (iiV.ofe)^v^ r \ ,3 O

Qs^ 6VMW-K* -'0,dfeS(^4>V^.flS 4 8o.fel)^V^) - « - 1 8 nS" .

The overal l "balance i s checked using the expression

' . • ? • . ' . . - . •

Q e == Qc •-*•' l Q 5 ^ (1 .11)

For Test 2 in BTU/hr

252.50 2 235.55 - 3.10 + 1.30 + 22.18 , or

252.50 = 255.93, l.kio off .

k. Heat Pipe Heat Transfer Rate (Q )

The heat pipe heat transfer rate is taken as the difference

"between evaporator coil power and the loss to the surroundings from the

evaporator section, i.e.

Q e ^ Q E - Qs^BvAWRiot*. (1.12)

For Test 2 in BTU/hr

5. Accuracy

The least accurate numbers used in heat rate data reduction are

the film, coefficients "between insulation and room air. The effect of

this uncertainty is exhibited in the overall heat balance of Equation

(l.ll). Typical data indicate the "balance is in agreement to "within

+ 5 Ver cent. For a typical heat rate of 300 BTU/hr., this represents

an uncertainty of + 15 BTU/hr.

APPENDIX J

HEAT SINK TEMPERATURE DATA REDUCTION

:A sample .calculation is provided for the sink temperature of

Test 2 (Chapter IV, Table 7). The sink temperature (T ) is taken as

the arithmetic average-of inlet and exit cooling water temperatures,

hence

T = T. • _ . + 0:. 5 ( $ T . cooling water), c inlet .-. ; /7

T. n . = 68.24°F (from method of Appendix H), inlet '

T = 4.37 (from method of Appendix G),

T = 68.24 + 0.5(4.37) = 70.43°F.

180

APPENDIX K

CONDENSER EXTERNAL UNIT CONDUCTANCE DATA REDUCTION

Evaluation of the unit conductance "between the heat sink and the

outer wick wall (t - YV/) is accomplished "by dividing the heat pipe

heat transfer rate "by the product of heat flow area and the temperature

difference "between wick wall and heat sink. The expression is

(?e = u c ( tTvr*u} ( T w - T ^ . (K.I)

Using the radial heat flow data reduction model of Appendix S, Equation

(K.l) is re-expressed in terms of measured parameters as

uc(ziTrwU) "- Tp..Tfc_ + u ( ^ O e •LtrXf L<.

(K.2)

where

T is the average measured temperature along the outer pipe

shell wall,

y is the thermal conductivity of the pipe shell material, and

L is the length of heat transfer along the condenser section.

Solving Equation (K.2) for U gives

l»(S c=[H^*[*>"- J0( 5£

-Y-l

(K.3)

181

For a sample calculat ion for Test 2 (Chapter IV, Table 7), the

parameter values a re :

Qe z 1 3 0 . 3 t S g (from Appendix I ) , W*

Tp ~~ 85 .57 °F (from Appendix E, with methods of Appendix H and Appendix Q),

Tc ~ "70.Mr3 DF (from Appendix J),

QT-il

X P - "S.1^ "wCrT R (from method of Appendix Q),

• p : 0.8IH- (from Appendix Q), AC

r = 0.375 inches, p

r = 0.326 inches, and "W

jj =5.00 inches.

Substitution of these values into Equation (K.3) gives the measured

value of the condenser external unit conductance in BTU/hr ft R as

Uc, [^VWH<^ or

BTU U c = M ^ . 8 1 VV*FTl°R •

182

APPEEDIX L

ESTIMATION OF INTERFACIAL INERTIAL FORCES

The pressure change across an evaporative interface with inertial

forces included is given by (l -)

p p - — ~ ^ U ^ (\-£A • , - x rv-Vx- v> — j r ~ V fx) ) or ( F : I )

(Pv-RiJ - Tm .. M i<r p„ ^ ftJJ ^ (W.2)

Using the pore radius, r , as an estimation of magnitude of the meniscus

radius , r , estimating u from Equation ( i l l . 2 ) , and using typ ica l

experimental data taken in th i s invest igat ion, the r e l a t ive magnitude of

the i n e r t i a l term i s calculated.

For water a t 100°F, Q = 300 BTU/hr, the estimation i s

Qt> .300 fH^L^Y LBm

P v ^ V ~ -2_-[T Vv L e W " (^TT^)^8)( l> ) ( \0^-?.Z^) ^ f c 00 / CTT- sec

f\, U v r 3.IO * \ o * "FFiic

f Vc \ (Pv U*£ / _ PA >0O^^ ( 3 . 1 0 ^ 0 ^ / ( ^ O . ^ A / J _ _ ? \

or

/ J C N M S A >A _ , ««vo-8

v^-j fv 1 p j »

183

hence the magnitude of the inertial term is negligible compared to

unity.

184

APPENDIX M

ESTIMATION OF VAPOR PRESSURE DROP

The relative magnitude of the vapor axial pressure drop is

estimated using Cotter's (8) Equations (II.6) and (ll.7)> typical water

data taken in this investigation, the meniscus pressure term

and Equation (II.30) for mass flow rate. This gives

\ f % / U v WV

(Mil) I I / \ / O ^ v WVv_\ / •= „ \ \ w-L

O e _ 1 m v - y , CKXSA

1M ~ MM

For water at 100°F, Q = 300 BTU/hr ^

K 4 ) * M (M.2)

|°v vAs; TV

-H- L«W\ Q e 3 Q Q (>H-H

P^ -SrrvLlKv/jL " ^X^aX^O0 3 '7 ' '2- (?*>0^ "" U 5 S M O F ^ S ^ ;

^ 5 5 ^ 0 " ^ ) ^ 8 ) / ^ o.s*8 ;

TTfv Yv* W " ( J ^ (350.4-y1 ( . 1 1 8 ^ 0 ° ^ - ^ 0 ^ o o X 3 Z ^

8/tCv Q e , „ -5

•vcp* rf W Z.16 *\o~5 ps\/ii*tv\ ;

i?<\ ^(ia~)^..i6*io-5)(\H''^(-5''*)-(-0't-08)(-^8j' = 5.-jfe«KfVrn ,

r<_ ^.oozis")

hence the vapor pressure drop is negligible.

185

APPENDIX N

EVALUATION OF WICK FRICTION FACTOR

Darcy's law for liquid flow in porous media (27) expresses the

pressure gradient as

' ^ ~ K r /WW. , (N . . I )

where K, the friction factor, is experimentally determined. For a

porous media consisting of tightly packed layers of 100 mesh screen

(O.Q045 inch wire diameter), Kunz, et al. (lT) report the experimental

value of

K -- -CoA *\0 8 '/FT* . (N>2)

For a wick, as employed in this investigation, with two layers of 100

mesh screen separated "by a liquid gap, the value of the wick friction

factor, K , is different from that given "by Equation (N.2). Since K is

"based on the area through which the liquid flows for tight wicks, K., is

also "based on the liquid flow area. From Figure N-l, the relation is

described "by

/ \ Ajt v^c*- —

^v " VN* N**,c* V/V X W , (N.3)

where A. . , is the cross sectional area of liquid flow in the wick t wick

and A„ is the cross sectional area of liquid flow in the gap. I gap j. & •*

Using the data from Figure N-l, Equation (N.3) reduces to

NOTE.: OJWeMSJONS IN MILS (TWOV$ftNCm\S OF /\N \NC\C)

FROM GEOMETRY :

k UQuiD , VMlCfc ~ 5 ^ . 13 M\Q-

AvUQU\0,<»W =• 8 0 . 0 0 VMl>

ATOTAL =• 18O. OO JVMI?

A U Q U \ P ToTA*-6 - PvREA. POROS\TY - — r -

/\T<>TfM-

11^21 = 0 w zSO , T 1 1

Figure N-l. Wick Cross Section for Evaluation of Wick Friction Factor.

r

K, ~- <>.!*\o+8) (f£tt) 7< FT7- } o r

188

This is the value of wick friction factor used in this investigation.

APPENDIX 0

ESTIMATION OF LIQUID HUERTIA TERM

The pressure drop term due to changes in inertia of axially

flowing liquid is given "by

JATOTAL

" V f = \ Q » t i i ) ) h , uA*er« ( o . D

_Sk— rLY ^ - ft^A-W UJ , *

or

Q uu

( 0 . 2 )

(0.3)

_¥£ / A pA e frwW \ M , tor (o.k)

J.C.- < "^ £ J U *• A A > ax\Jl " ( 0 . 5 )

__& r it+u+u-i\ ^ ^ W A \ i ^ y ;.for (o.6)

JL*4* £ \ <• Ac*JU* ie . (0.7)

These velocity relations are based on the model of uniform injection

and rejection of liquid along the condenser and evaporator sections

respectively. Equations (0.2), (0.^-), and (0.6) are simplified "by

ou ( . W X •¥($ a « e (0-8)

190

-

Qe

and T(V) is a linear function, piece-wise continuous. Substituting

Equation (0.8) into (O.l) gives

fXXOTfr'-

(0.9)

The integral in Equation (0.9) is evaluated piece-wise:

for 0 6 ^ . £ '-J£c

^ V ) -- ^ ; . •

taxxv> - ^ ,

for | c ^ ^ 4 Xc + I * ,

V(V> ^ l , •

fJL+K

) l c l 0 ^ A> = 0 ;

for • JL+Jk £ ^ * JL + JU + ie.,

VIS) "- ^ ,

hence n (-ToTfc*- i i . 1 . . « - « _ o ^ ^ <H = ^ 0 - ^ = O > a r

the net change in inertia is zero. This could have "been deduced from

the fact that the initial and final velocities are zero, hence no net

change in inertia.

Comparison is made of the relative magnitude of inertial to

frictional pressure drop only over the condenser section:

fe;igfd_- • • (o.xo)

[^-fe;(^) UPLECTA. _ Q*

W W - " «-^WK,Al^ * (0.11)

Equation (O.ll) is.evaluated using the wick parameters of the investi

gation and conditions of water at 100 F, Q = 300 BTU/hr. Equation

(O.ll) is evaluated as

j^W,* __ __?^. -<p2l£---\

yx\\»**o* _ ^^v y.io's

\ ^ M FfcicTtOM

hence the inertia term is negligible compared to the frictional term.

192

APPENDIX P

ESTIMATION OF MAGNITUDE OF THERMAL RESISTANCES

From Equation (V. 8) the total thermal resistance is

RT^TAL =' R e * + • Re> -*• fcv -v Re; -v fccw . (P.l)

Dividing by the condenser wick term provides a nondimensional comparison

-o ^ "V ^CVN ^ Rcw He* . (P-2) lSCVY

The wick resistance magnitudes are estimated "by the conduction model of

JnC^L fco* = To£ff~Jtc } OLnd ( P . 3 )

The vapor resistance magnitude is estimated "by using the Clapeyron

equation and the estimated vapor axial pressure drop

p\ Vw fcT ) - ^ T X T T N T (p.5)

/SAT. ^ V > ft) ) ^ ^

193

where

p = w ( ^ # ) ( ^ ^ - ^ y (p.6)

A/iAwVv Y - """/XT' ; *n4 (P.7)

_5> " % - = PwV^TVY-vA . (P.8)

Substituting Equations (P.6), (P.7)> and (P.8) into Equation (P.5)

gives

_AX. 8 Ay TO JU>Tfti/v _ j ^ V 3 \\ v ^

Rv^ Qe = ^ r ^ M ^ r v ^ p j ^ ^ " ^ ^ . (p.9)

The interfacial thermal resistances are estimating using the model

theory of Wilcox and Rohsenow (37) which is an extension of the Schrage

(30) theory and with an accumulation coefficient of unity. This model

gives

(£k .i/U—S j\™ *t T«% M ;

(p.10)

R<u = 7 -T^rXjl-RlSr-J , (Pai)

where T and P are the saturation temperature and pressure respect

tively, and R is the gas constant of the vapor.

Using typical data taken in this investigation for water at

100 F, Q = 300 BTU/hr, the ratios of thermal resistances are estimated

as

19

-*r~s Te = L > (p-12)

fc* \(o Av/^oilcd^a^y^f/ P, *V \k Av/'Vo^d^a^) Xtff^, ?A/ 3v " ^ x „ ,*4 ,

-s-^- ;r.-- , v — - 3 . « M O * a*& (P.14)

R.c« " -2-rv\vNC^R, Vi^ ~ ' • (p"15)

Hence the wick resistances are retained and the vapor and interfacial

resistances are neglected.

195

APPENDIX Q

EVALUATION OF CONDENSER ACTIVE LENGTH FROM MEASURED PARAMETERS

Evaluation of the measured active length of condensation (Lp) is

accomplished "by use of a data reduction model "based on l) radial heat

conduction through the wick and pipe shell, 2) measured vapor and outer

shell wall temperatures, and 3) "the measured heat pipe heat transfer

rate. A, sample calculation is. provided for Test 2 (Chapter IV, Table 7)

For radial conduction of heat from inner wick surface (r = r ), v

through the outer wick surface (r = r ), and to the outer pipe shell w

surface (r •= r ), the heat flow expression is ,

T„ - TP .

Q* = ' JnC&L + M l L (Q-X)

Solving for the value of the condenser active length ratio gives

Qfu (_^L-\r j-iSL + po _wM i (Q2)

where Q. is the measured heat pipe heat transfer rate,

T is the measured heat pipe operating temperature (equal to the

vapor and adiabatic wall temperature),

T is the average measured temperature along the condenser at the

outer pipe shell wall (r = r ); in terms of thermocouple num-ir

bersi

196

TP - T S [ * ( T C - H } * ^ C T C ' 0 + i("tc-e) + (Q*3)

-V 3 (TC- 2G) 4- z (~xc-in) + i (Tc-i8)l

y is the thermal conductivity of the pipe shell material,

evaluated at T , o

Xj is the thermal conductivity of the liquid working fluid

evaluated at T , and o

Xp/X«ffis the thermal conductivity ratio of liquid to effective

evaluated at T and using Equations ("V. 19) and (V.20), see

Appendix R.

For Test 2, the parameter values are:

Q = 230.32 BTU/hr (from Appendix i),

T = TC-14 = 102.3T°F (from Appendix E, with method of

Appendix H),

T = 85.57 F (from Equation (Q.3)> using data from Appendix E, ir

with method of Appendix H),

X =8.42 BTU/hr ft R° (for 30^ stainless steel using data of

Reference (±6)),

Xjg = 0.3621 BTU/hr ft R° (for water using data of Reference

(ho), methanol ( 3))>

*£ = 0.5593 (from Equations (V.19) and (V.20), with the wick

solid material as 316 stainless steel using data of

Reference (l6)),

r = 0.375 inches, sr r = O.326 inches, w

r = O.298 inches, and v

9 =5-00 inches.

Substitution of these values into Equation (Q>2) gives the value of the

condenser active length ratio as

_U JJo:32L__ [ hS^l^^fo^^^^^ it ' (jOZ.VJ-e'S.^) L'^Cs.H^XO '2-tr 0.3 iv)(5') J or

^ = 0.81* .

198

APPENDIX R

DERIVATION OF EQUATION FOR EFFECTIVE WICK THERMAL CONDUCTIVITY

The effective thermal conductivity of the wick in this

investigation depends on. the wick geometry and the conductivities of

the liquid and wick-solid materials. An expression for the effective

conductivity is derived in two steps l) the effective conductivity for

a single layer of screen, ~^w, is modeled "by combined series and parallel

conduction, and 2) the overall wick effective conductivity, X«$f; is

modeled "by series conduction through two layers of screen separated "by

a liquid gap.

A. Thermal Conductivity of a Single Layer of Screen (X^Y

The wick is modeled "by considering a typical section of screen,

see Figure R-1. Each section is composed of liquid and/or solid. The

conduction area for each section is different, "but each total length

(screen thickness) of conduction is the same. Defining the thermal

resistance as

R length of conduction / >. "~ (thermal conductivity)(conduction area) ^ * '

and using the geometry indicated in Figure R-1, the respective resis

tances are

H- r*$ 2.rc--2.Yws

199

Jfo* ^ rc rc ^

n \ ^ \ J?"

a

V*

1 i 2 " 1 3 a

k*

I

"

1 • • • • . I S\\ 1 1

Q +

1 i

= * •

1 r

i

A-

,SOHO

i

A-

• LIQU\D

f SIDE V1E.W

SECTION I

PLfcN V\EW ^ r c

i

SOUO

Z

I3

+ z

£ ^

LIQUID

pj

1 '

SIDE VIEW SECTION Z

S\DE VlE-W

SECT\ON 3

Figure R-1. Geometry Model for Conduction Through One Layer of Screen.

200

XJIA ****"«• } W (R.3)

j^Or^j^O,, ^% - y* . (R.^)

Combining these resistances in parallel gives

P.VA/ R» Rn. K, ?

where R is the thermal resistance of one layer of screen given "by w

R~ = ~xT(? (.r^tiff \ (R-6)

Combining Equations (R.2) through (R.6), and expressing in nondimen-

sional form gives •_

% - TJ^OPMS^T+ ww^)+ (R'T)

+

B. Effective Wick Thermal Conductivity

The two layers of screen and their separating liquid gap, see

Figure R-2, form a series of concentric cylinders. Defining the thermal

resistance as

i /outer radius\ R __ _ _ ^ Vinner radius/ / «v ~" (2ir)(thermal conductivity)(axial length) ' \ • J

the respective resistances of each annulus is

\r\ ( r*- - CApta*^ )

z» = -—~^^i ~ i (E-9)

201

Figure R-2. Annular Cross Section for Conduction Through Screen Layers with Liquid Gap,

202

/ rw^^O^-^X

*» -- - — ^ 7 T > • " " ' (R-10)

Tv/ +'t (yv/s+r^)

^ c = • i^ZJ • (R-I:L)

Combining the resistances in series gives

R ^ff - £ * +• Rs -v £c (R.12)

where R „„ is the effective thermal resistance of the wick given "by err

In ("£)

*•« -- ~^j^rr . (R-13) Combining Equations (R.9) through. (R.13) and expressing in nondimension-

al form gives

XJ?

X^F ^ (|¥)[U (^t^) + (R-in

r^^c^v-cCv+^y"1 + lhv W * M T T ^ W ^ ^

where n is the number of screen layers (two for this investigation)'

forming a wick with one layer of screen at the outer annulus, a liquid

gap at the second annulus, and other screen layers constituting the

inner annulus.

Equations (R.T) and (R.1*0 together express the effective thermal

conductivity of the wick used in this investigation.

203

APPENDIX S

EVALUATION OF EVAPORATOR ACTIVE LENGTH FROM MEASURED PARAMETERS

Evaluation of the .evaporator active length from experimental data

is "based on a data reduction model in which the pipe shell is a fin with

l) axial heat flow within the fin, 2) heat addition along the outside of

the fin from the evaporator heating coils, and 3) heat rejection along

a fraction of the inside of the fin to the heat pipe wick (see Figure ~

S-l). For this model, the necessary experimental data include the heal

ing coil power, the heat loss rate to the surroundings from the evapo

rator, the heat pipe operating temperature (vapor temperature), and the

temperature at an axial location along the designed evaporator length.

The equations necessary for this data reduction model are developed and

solved, and a sample calculation made for Test 2 (Chapter IV, Table 7).

1. Development of Expression for L

The energy "balance for the differential control volume of length

dx shown in Figure S-l is described "by the diagram of

2ir V«ff U*)

( - X f N ^ +

(rto - TJ) H&ti

TxJ ~*M^

%.(<»*)

DES\G»N

Figure S -1 . S h e l l Diagram for Evapora tor .

;,ro ©

where

% ^ x* , ««* (s-1}

Ap - ^(r^-Tvi) . (s.2)

Balancing these energy terms gives

-vr*r&)+M-**&)**+ I3§£V(T O-T :) - ^

V%(^ -**r£ . Cancelling terms and assuming constant pipe shel l thermal conductivity

gives

£ 1 , I* Xett „ i l l , ^ Ke« / _ . . __\ Q£

or (SA)

11 T I R? - r f Rf i - -T j . I o A ~ l

J^- i?x: T = - \_T^eT o +u^Q eJ ; (s-5

where

A Xf ftp ;

^? - ~vTXo c^a (s-6)

R* ^ ^ (r^ja l . (S.T) IT»"Y«« JU

The boundary conditions for Equation (S.5) are

B.C. 0 y = o ; - X p A p ^ = Q E i e " ^s.e ) (S.8)

206

where § is the heat rate from the evaporator section to the s,e

surroundings (see Appendix I),

B.C. i) *= L U 5-L } T - "T (s.9) )

"where T is the temperature measured "by thermocouple numher 113.

(TC-113), and

&-c- V x- L e ; a* " u ; (s.io)

i.e., no further axial conduction along the shell. Three "boundary

conditions are required for solving Equation (S.5) since Equation (S.5)

is of second order, and the origin of the coordinate (describing the

position of L ) is unknown.

The general solution to Equation (S.5) is

T(V) - C, COSH (\**) + C^ 5INV\ (>*) + T0 + fc*Q£; (S.ll)

where

A x= t . c%y.. -

Substitution of the "boundary conditions, Equations (S.8) and (S.9) gives

the solution of

T ( ^ r L^TsH(x(u^r + k<W<fe/)* (s.13)

A 5INW ( x ^ ^ - L ) ) y

\ ^ i ; ' O s ^ ( x x ) -V- To + RwQe .

Substutition of the "boundary condition of Equation (S.10)., Equation

(S.12), and the identities

Le + L - A aTk«

rives the solution for L as 5 e

(S.lV)

(COSH A ] (cos* B j - U>M* /V) f$\*i* 5J - COSH(A-B) (S.15)

JU Q t LQe(rf I" bL^ s - ( ( $ ^ ^ * « ;

COSH feto-^) (s.16)

The position of thermocouple number 113 is given "by

Ln3_ \

JU " * .

For typical heat pipe operation in this investigation,

Le Ti > o. 5 , ^

YL

;^ j JU > l ° ,., ^ c e

(S.1T)

(S.18)

(S.19)

\%& U

(S.20)

hence the "bracket terms may "be approximated as unity. This approxima

t ion , and Equation ,(S.l6) gives

208

2A - A-<bA _ f ^ ^ - GAM J$&-^ (s.21)

f* - r ^ r -i^o^t w c Equation (S.2l) is the expression used to reduce experimental data to

obtain the evaporator active length ratio.

2. Sample Calculation

For Test 2 (Chapter IV, Table 7) the values of the variables are:

Q =22.18 BTU/hr (from Appendix I), S y 6

Q j = 252.50 BTU/hr (from Appendix I),

T ' '= 102.3T°F (from Appendix E with method of Appendix H),

T = 210.90°F (from Appendix E with method of Appendix H),

i> = 3-36 inches,

r = 0.326 inches w

r = 0.298 inches, and v '

r = 0.375 inches. P •

5e The value of the evaporator wick thickness parameter f -y is

approximated by using Equation (V.33); re-written as

where the function f, is plotted in Figure 2k for parameter values used

in this investigation (see Appendix T). The meniscus radius ratio,

Equation (V.k6), re-written as

209

Is first evaluated with; the additional parameters of

r = .00275 inches,

8 2 K = 2.6l x 10 l/ft (from Appendix N),

Q = 230.32 BTU/hr (from Appendix I), and

f'^J = T.9T6 X 1010 BTU/hr ft2,, where "Mjt

ft = 61.966 l W f t 3 (for water (4o), methanol (33) )>

CT = 4.778 X lO-3 Ibf/ft (for water (42), methanol (42)),

h . = 1035.7 BTU/lbm (for water (4o), methanol (33)).*

M* = 138.4 x 10"^ lbf/sec/ft2 (for water (4o), methanol (4l)).

Substitution of these values into Equation (S.23) gives

^ - 4-.IO .

From Figure 24, a t = 45 ,

- i * - = o.S<b5 . Tv/ - TV

Hence the remaining parameters are

S«> = 0.865(.028) = 0.02422 inches,

Xf = 9-39 BTU/hr ft°R (from Reference (l6), at temperature of TC-113), and

X**f= 0.6474 BTU/hr ft°R ("by method of Appendix R, and References (4o) and (l6), at temperature T ).

The resistance terms are

JU_ w j£ RP - XPTr(r ?^r^ = 3<\.i<\i B T U ^

^(-YwTS^j m*-^

~Z*Y*HH< ~ O'0(O1 n 8 B r u ' a™

210

fV^-- r i M r ^ O

Substi tution of these values into Equation (S.2l) gives

L<2 o.zzo^ = o.qmfe —

Using Newton's method of approximation for an iterative solution gives

T; = °-1"0

211

APPENDIX T

DERIVATION OF RELATION FOR EVAPORATOR EFFECTIVE WICK THICKNESS

The retreat, in the radial'direction, of the liquid-vapor

interface into the wick of the evaporator section is modeled by a

spherical surface, radius.r , attached, "with contact angle <$>, to a solid

torus, inside radius r and solid radius r , see Figure T-l. The ' C W S ; . to •

effective thickness is taken as the height from the bounding surface

below to the minimum point of the spherical surface. For given values

o f r , r , r > r , and d>, the thickness Se is implicitly determined v w vs m '

with the aid of the generating angle <X. From Figure T-l, the geometri

cal relations are

Y~m COS (oL+ 4>^J -V- f w s COS c* = Tw/s -v T t } o-Yvi ( T . l )

S G = (Vw-rJ) -2

Expressing in dimensionless form gives

\ + /n" (V^A COS ( oL -v <^^

, a™l (T.3)

&)-- 0-^)-(^— - ( ^ ( - ^ V <*• * )

Eliminating the generating parameter c< by combining Equations (T.3)

and (T. -) results in the function form of

212

PARAMETER VM-UG5 (THIS INVE.STVG,/ \T\ON)

y^ = o . 1Z(O '*• rv ~ o . z ^ e ^. f -z. O. OC2.T5 »rv.

Figure T-1. Interface Model for Derivation of Se

f $* \ i f r« s» Ji. H A /m \ \P*) ~ fi^lv, Ti, lv ,.rc,*) . (T-5)

From algebra and simple trigonometric identities, the explicit form of

S^ i s found to be

iTw - T v (fc-v) l

. r , *k_( jaa. ^ J ^ cos <*Y

4- (T.6)

- r SIM' %(^+Jfe°**T ( ¥v-0 £•£«- • )

£ - ^ sw4> V")1

Y V L

rv - i - {(^+^c«*T+fe""?},t

fe_%w*Y +VMMili^)N IV _ i Iv

TvJ

~?5 - \

L ~~ Qfe-0 C H J JJ J ' Sg v^

Chapter V, Figure 2k indicates a plot of tw-Yv vs. - - for the

values of r , r , r , and r used in this investigation. c ws w v

2l4

APPENDIX U

EVALUATION OF INTERNAL THERMAL RESISTANCE

The internal overall thermal resistance is given "by Equation

(V-68) as

Tew~Tcw R ^ : " , (U.l)

Qe

where *Tew and Tiw are the average temperatures at r along the active w

lengths of evaporator (L ) and condenser (L ) respectively. Evaluation

of measured and predicted values of the resistance R, used in Figure 3^->

is accomplished "by substituting measured and predicted wall temperatures

into Equation (U.l). A sample calculation is made for Test 2 (Chapter

IV, Tahle 7). •

1. Evaluation of R from Predicted Parameters

The predicted values of the average wall temperatures are evaluated

from simple conduction models through the active lengths of the wick.

For the condenser,

\*C-&)_ TCw = T0 - Qe. T^X^^c . or (u.2)

For the evaporator,

Te„ ^ To +

. *

Tew . - " T c + Qe

For Test 2 of Tahle 7, the solution of the system of Equations (V-46)

through (V-55) vith the independent parameter values of

Q = 230,32 BTU/hr, :' e

U = 234.89 BTU/hr ft,2°R,

T = 70.43°?,

gives the predicted parameter values of

T = 102.55°F, o '

y s = 0.3622 BTU/hr ft°R,

I T = 0-5593,

5« = ( V rv^[^l = (°-028)C°-865) = 0.02422 inches,

u

X = 0.809.

W ( Xs,*) Q* T T w U , or ^

wC f rJ) fi*\/ JU\ ^ xA i l \*«ff) I L«)

216

Substitution of these values into Equations (U.3) and (U.5) gives

T = 8j„kj°F, cw

T = 12'3.55°F. ew

Substitution of these values into Equation (U.l) gives the predicted

value of thermal resistance as

R = 0.1567^j? •

2. Evaluation of R from Measured Parameters

The measured values of the average wall temperatures along the

active lengths are evaluated from simple heat transfer models based on

thermocouple measurements of the shell temperature. For the condenser,

=r — + A W ( ^ Tew - V + <*« -^TK^Lc , or (U.6)

U(%1 (i<

where T is the average measured temperature at r = r along L . For p to p to c

the evaporator, the data reduction technique of Appendix S is used to

evaluate T from measured parameters. This average temperature is

given by

J _ (Le , Tew = L€ \ T(*) <** , (U.8)

^o

where T(x) is given by Equation (S.13). Substitution of Equation (S.l6)

into Equation (S.13) and evaluation of the integral of Equation (U.8)

gives

217

»€w "~ ' \l"i ~" KvJ Qg ?*«V, C?s^ H f f U V K £ f i 2 ! ^ ^ - , -ill (U.9)

where T .. Is the temperature measured by thermocouple number 113. For

Test 2, Table J, the term values based on measured parameters are

T? - 85.5T°F (from Appendix Q~),

Xf = 8.42 BTU/hr ft°R (from Reference (l6)),

HT = 0.8l4 (from Appendix Q), .Xc

Q<» = 230.32 BTU/hr (from Appendix I),

r T„* = 210.90°F (from Appendix E with method of Appendix H),

l* = O.06TT8 hr°R/BTU (from Appendix S),

E? = 39-792 hr°R/BTU (from Appendix S),

Qz = 252.50 BTU/hr (from Appendix I),

Qs,e= 22.18 BTU/hr (from Appendix I), and

^f- = 0.7^0 (from Appendix S).

Substitution of these values into Equations (U.7) and (U.9) gives

T = 87.37°F, C ¥ •

T = 123.92°F . ew - y

Substitution of these values into Equation (U.l) gives the measured value

of thermal resistance as

R - O.I 5 87 - ^ m

218

APPENDIX V

DATA AND THEORIES OF OTHER INVESTIGATIONS

1. Kunz, et al,,,} Reference (17)

A calculation is made for Kunz's theory on capillary limited

maximum heat transfer rate for the materials and geometries used in this

investigation. From Equation (ll.33)>

VwWXO^") lf

Q^X. - ^ Ai' j V K^JUf^U*. - (V*1)

For this investigation, typical parameter values are:

PjL 0 - W _ R 9 .-JO _BTiJ

><je

(for water a t 100 F, see also Appendix U),

b= .Tr(Yw*-rv) - ir (,326-i-/t<u J - l . U o ^ m . ;

S = Tw-Tv = .3ife. - . ^ 6 = o . 0*2.8 tn. /

J^v -z Z . 6 I *IG* /FT7- ( see Appendix N),

Tc .O 0*2.15

rm,w. = cos> ~'~lyioJy' '- o.oo*aQ<n m. .

Using these values gives

QmA* = 6W.S -£*"

2. Sim and Tien, Reference (3 -)

A calculation is made of the Sun and Tien parameters which indi

cate the presence of axial heat conduction along wick and pipe shell.

From Equations (11.39) and (lI.4o)

if = : Af. / - ^ ^ — . anJ (V.2)

B = « * \ ^ ) . (v.3)

For this investigation, typical'parameter values are:

X ==. 18.0 inches,

Vj» = 0.375 inches,

tw =0.326 inches,

rv = O.298 inches,

"Xp = 8.52,BTU/hr ft °R (for 30 - stainless steel at 100°F, from Reference (l6)),

Xe«= 0.6 5 BTU/hx1 ft °R ("by method of Appendix R, and Reference (ko) and (16) at 100°F)/ and

H = 220 BTU/hr ft2 °R ("by methods of Appendix K).

Using these values gives

|\A -. \T-(o , oovi

B.x - o.q'si .

Since NA>loo and B.i <• 1 Sun and Tien suggest axial conduction is

negligible.

3. Schwartz, Reference (29)

The experimental work of Schwartz provides data for the indica

tion of variation in active condenser long the Schwartz did not refer

to an active length, "but concluded the effective wick thermal • ::•:-..--LI:.:-

220

conductivity to "be a variable. Schwartz's data is reduced to the

present correlation "by calculating the active condenser length. For -

condenser heat transfer, the active length is expressed "by

(VA)

where i s the temperature difference across the wick. The vapor ax ia l

Reynold's num"ber i s given, "by

p V ^ f O _3__ Qe

Schwartz's materials and parameter values are

working fluid: water

wick: 100 mesh stainless steel screen,

geometry: r = 0.199 in. w

r = 0.189 in. v

I = 2.8 in. . c

From Appendix R X*

T^ ~~ °'3°5 .

Using these values and Equations (V.4) and (V.5)j Schwartz's data is

reduced as:

test

1

2

T 0 A *T L /I c' c Re

V

°F BTU/hr F° ~ ~

98 6 l . 7 3.1 0.592 104.5

13^ 169.3 6.k 0.756 272.0

k. Miller and Holm, Reference (2k)

The experimental work of Miller and Holm provides data for the

indication of variation in active condenser length. Miller did not

recognize this variation.Miller's materials and parameter values for

the model heat pipe are

working fluid: •water

wick: 200 mesh nickel screen

geometry: r = 0.^12 in. w

r .= 0.352 in. . v •

I = 5.0 in.

From Appendix T,

^ = 0.061 .

Using these values and Equations (V.4) and (V.5)> Miller's data is

reduced as:

test T 0

Q *T L fl c c

Re V

°F

310

BTU/hr

62.8

F°

2.8

<v s+~t

1

°F

310

BTU/hr

62.8

F°

2.8 0.209 hk.5

2 323 . 68.0 2.k 0.264 kj.6

3 336 72. k 2.7 0.250 50.6

k 350 77.9 3.2 0.228 5^.1

5 363 83.3 2.7 0.292 57.8

6 380 90.5 3.0 0.288 62.7

222

5• Fox, et al., Reference (lo)

The experimental work of Fox, et al.; provides data for the

indication of variation of active evaporator.length. Fox did not

refer to an active length, tout, concluded that a vapor film was present

in sections along the outer region of the evaporator wick and liquid V

along the inner region. Fox's materials and parameter values are

working fluid: water

wick: 100 mesh stainless steel screen

geometry: • r'< = 0.722 in. • v •

r = 0.672 in. v

t = 6.0 in. c

Although the Fox data was incomplete to permit correlation, Figure (V-l),

redrawn from Figure 2 of Reference {3^-) > does indicate a variation in

active length. The vapor film concluded by Fox would explain the mag

nitude of thermal resistance along the evaporator, but not the varia

tion of thermal resistance. From Figure (V.l) it is seen that the slope,

expressed by 1 • .

T* " A Trf , (v.6)

where A is a constant, does decrease for increased power levels,

indicating the active length does decrease.

r

223

l4-oo

\2oo -

QOO

Figure V-1. Power Versus Evaporator Wick Temperature Difference for Data of Fox, et al. (10).

22^

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228

VITA

Colquitt Lamar 'Williams was "born in Valdosta, Georgia,, on

June 20, 19^, the son of Mr. and Mrs. Walter H. Williams Jr. He

received elementary and high school education in Gulfport, Anona, and

Largo, Florida. He attended the Georgia Institute of Technology from

1962 to 1967 on the Co-operative Plan where he won the degree of Bach

elor in Mechanical Engineering in June, 1967. ^ June, 1968, he

obtained a Master of Science in Mechanical Engineering.and continued

work on.the Doctor of Philosophy degree at Georgia Tech. His financial

support in graduate school was "by an NDEA Fellowship and teaching

assistantships in thermodynamics and fluid flow.

He married Carrolyn Jo Lovering, a native of Lakeland, Florida in

January, 1973.